Invariant theory bcgimFrom Wikipedia, the free encyclopedia

Contents

1 Bracket algebra 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Bracket ring 22.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Canonizant 33.1 Canonizants of a binary form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Capelli’s identity 44.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Relations with representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.2.1 Case m = 1 and representation Sk Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2.2 The universal enveloping algebra U(gln) and its center . . . . . . . . . . . . . . . . . . . 64.2.3 General m and dual pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3.1 Turnbull’s identity for symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3.2 The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices . . . . . . . . . . . . 104.3.3 The Caracciolo–Sportiello–Sokal identity for Manin matrices . . . . . . . . . . . . . . . . 104.3.4 The Mukhin–Tarasov–Varchenko identity and the Gaudin model . . . . . . . . . . . . . . 104.3.5 Permanents, immanants, traces – “higher Capelli identities” . . . . . . . . . . . . . . . . . 11

4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Catalecticant 145.1 Binary forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Catalecticants of quartic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Cayley’s Ω process 16

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6.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Chevalley–Iwahori–Nagata theorem 187.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Chevalley–Shephard–Todd theorem 198.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Differential invariant 219.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10 Evectant 2310.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11 Geometric invariant theory 2411.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.2 Mumford’s book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

12 Glossary of invariant theory 2812.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.2 !$@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.5 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.6 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.7 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.8 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.9 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.10H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.11I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.12J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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12.13K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.14L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.15M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.16N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.17O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.18P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.19Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.20R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.21S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812.22T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.23U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.24V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.25W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.26XYZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.27See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.29External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13 Gram’s theorem 4213.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14 Gröbner basis 4314.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

14.1.1 Polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.1.2 Monomial ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.3 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.3 Example and counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.4 Properties and applications of Gröbner bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

14.4.1 Equality of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.4.2 Membership and inclusion of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.4.3 Solutions of a system of algebraic equations . . . . . . . . . . . . . . . . . . . . . . . . . 4714.4.4 Dimension, degree and Hilbert series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.4.5 Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.4.6 Intersecting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4.7 Implicitization of a rational curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4.8 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4.9 Effective Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.4.10 Implicitization in higher dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

14.5 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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14.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

15 Haboush’s theorem 5415.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16 Hall algebra 5616.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

17 Hermite reciprocity 5817.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

18 Hilbert’s basis theorem 5918.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

18.2.1 First Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.2.2 Second Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

18.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4 Mizar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

19 Hilbert’s fourteenth problem 6219.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.2 Zariski’s formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.3 Nagata’s counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

20 Hilbert’s syzygy theorem 6420.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

21 Hilbert–Mumford criterion 6521.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

22 Hodge bundle 6622.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

23 Invariant estimator 67

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23.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.1.2 Some classes of invariant estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.1.3 Optimal invariant estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.1.4 In classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

23.2 Mathematical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.2.3 Example: Location parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.2.4 Pitman estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

23.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

24 Invariant of a binary form 7124.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.3 The ring of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

24.3.1 Covariants of a binary linear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.2 Covariants of a binary quadric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.3 Covariants of a binary cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.4 Covariants of a binary quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.5 Covariants of a binary quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.6 Covariants of a binary sextic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.7 Covariants of a binary septic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.8 Covariants of a binary octavic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.9 Covariants of a binary nonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.10 Covariants of a binary decimic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.11 Covariants of a binary undecimic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.12 Covariants of a binary duodecimic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

24.4 Invariants of several binary forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.1 Covariants of two linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.2 Covariants of a linear form and a quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.3 Covariants of a linear form and a cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.4 Covariants of a linear form and a quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.5 Covariants of a linear form and a quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.6 Covariants of a linear form and a quantic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.7 Covariants of several linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.8 Covariants of two quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.9 Covariants of two quadratics and a linear form . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.10 Covariants of several linear and quadratic forms . . . . . . . . . . . . . . . . . . . . . . . 7424.4.11 Covariants of a quadratic and a cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.12 Covariants of a quadratic and a quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.13 Covariants of a quadratic and a quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

vi CONTENTS

24.4.14 Covariants of a cubic and a quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.15 Covariants of two quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.16 Covariants of many cubics or quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

24.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

25 Invariant polynomial 7725.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

26 Invariant theory 7826.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7826.2 The nineteenth-century origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.3 Hilbert’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.4 Geometric invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8026.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8026.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

27 Invariants of tensors 8227.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8227.2 Calculation of the invariants of symmetric 3×3 tensors . . . . . . . . . . . . . . . . . . . . . . . . 8227.3 Engineering application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

28 Kempf–Ness theorem 8428.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

29 Kostant polynomial 8529.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8529.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8529.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8629.4 Steinberg basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8829.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

30 Littlewood–Richardson rule 9030.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

30.1.1 Littlewood–Richardson tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9130.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9130.1.3 A more geometrical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

30.2 An algorithmic form of the rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9230.3 Littlewood–Richardson coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9230.4 Generalizations and special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

CONTENTS vii

30.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9530.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

31 Modular invariant theory 9731.1 Dickson invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9731.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9731.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

32 Moduli space 9932.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

32.2.1 Projective space and Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.2.2 Chow variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.2.3 Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

32.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.3.1 Fine moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.3.2 Coarse moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10132.3.3 Moduli stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

32.4 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10132.4.1 Moduli of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10132.4.2 Moduli of varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.4.3 Moduli of vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

32.5 Methods for constructing moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.6 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

33 Molien series 10533.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

34 The Classical Groups 10734.1 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10734.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10734.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 109

34.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10934.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Chapter 1

Bracket algebra

In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra witha symbolic representation of projective invariants.Given that L is a proper signed alphabet and Super[L] is the supersymmetric algebra, the bracket algebra Bracket[L]of dimension n over the field K is the quotient of the algebra BraceL obtained by imposing the congruence relationsbelow, where w, w', ..., w" are any monomials in Super[L]:

1. w = 0 if length(w) ≠ n

2. ww'...w" = 0whenever any positive letter a ofL occursmore than n times in themonomial ww'...w".

3. Let ww'...w" be a monomial in BraceL in which some positive letter a occurs more than n times, andlet b, c, d, e, ..., f, g be any letters in L.

1.1 See also• Bracket ring

1.2 References• Anick, David; Rota, Gian-Carlo (September 15, 1991), “Higher-Order Syzygies for the Bracket Algebra andfor the Ring of Coordinates of the Grassmanian”, Proceedings of the National Academy of Sciences 88 (18):8087–8090, doi:10.1073/pnas.88.18.8087, ISSN 0027-8424, JSTOR 2357546.

• Huang, Rosa Q.; Rota, Gian-Carlo; Stein, Joel A. (1990), “Supersymmetric Bracket Algebra and Invariant The-ory”,Acta ApplicandaeMathematicae (Kluwer Academic Publishers) 21: 193–246, doi:10.1007/BF00053298.

1

Chapter 2

Bracket ring

In mathematics, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d by dminors of a generic d by n matrix (xij).The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plückerembedding.[1]

For given d ≤ n we define as formal variables the brackets [λ1 λ2 ... λd] with the λ taken from 1,...,n, subject to [λ1λ2 ... λd] = − [λ2 λ1 ... λd] and similarly for other transpositions. The set Λ(n,d) of size

(nd

)generates a polynomial

ring K[Λ(n,d)] over a field K. There is a homomorphism Φ(n,d) from K[Λ(n,d)] to the polynomial ring K[xi,j] innd indeterminates given by mapping [λ1 λ2 ... λd] to the determinant of the d by d matrix consisting of the columnsof the xi,j indexed by the λ. The bracket ring B(n,d) is the image of Φ. The kernel I(n,d) of Φ encodes the relationsor syzygies that exist between the minors of a generic n by d matrix. The projective variety defined by the ideal I isthe (n−d)d dimensional Grassmann variety whose points correspond to d-dimensional subspaces of an n-dimensionalspace.[2]

To compute with brackets it is necessary to determine when an expression lies in the ideal I(n,d). This is achievedby a straightening law due to Young (1928).[3]

2.1 See also• Bracket algebra

2.2 References[1] Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Ency-

clopedia of Mathematics and Its Applications 46 (2nd ed.), Cambridge University Press, p. 79, ISBN 0-521-77750-X, Zbl0944.52006

[2] Sturmfels (2008) pp.78–79

[3] Sturmfels (2008) p.80

• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525, Zbl 0196.05802

• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102, Zbl 0258.14011

• Sturmfels, Bernd (2008), Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation(2nd ed.), Springer-Verlag, ISBN 3211774165, Zbl 1154.13003

• Sturmfels, Bernd; White, Neil (1990), “Stanley decompositions of the bracket ring”,Mathematica Scandinavica67 (2): 183–189, ISSN 0025-5521, MR 1096453, Zbl 0727.13005

2

Chapter 3

Canonizant

In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical formfor them.

3.1 Canonizants of a binary form

The canonizant of a binary form of degree 2n – 1 is a covariant of degree n and order n, given by the catalecticant ofthe penultimate emanant, which is the determinant of the n by n Hadamard matrix with entries ai₊jx + ai₊j₊₁y for 0≤ i,j < n.

3.2 References• Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. (2nd ed.), Oxford. ClarendonPress, JFM 26.0135.01

3

Chapter 4

Capelli’s identity

In mathematics, Capelli’s identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) =det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebragln . It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley’s Ω process.

4.1 Statement

Suppose that xij for i,j = 1,...,n are commuting variables. Write E ᵢ for the polarization operator

Eij =

n∑a=1

xia∂

∂xja.

The Capelli identity states that the following differential operators, expressed as determinants, are equal:

∣∣∣∣∣∣∣∣∣E11 + n− 1 · · · E1,n−1 E1n

... . . . ......

En−1,1 · · · En−1,n−1 + 1 En−1,n

En1 · · · En,n−1 Enn + 0

∣∣∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣x11 · · · x1n... . . . ...xn1 · · · xnn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂∂x11

· · · ∂∂x1n... . . . ...

∂∂xn1

· · · ∂∂xnn

∣∣∣∣∣∣∣.Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded withall terms preserving their “left to right” order. Such a determinant is often called a column-determinant, since it canbe obtained by the column expansion of the determinant starting from the first column. It can be formally written as

det(A) =∑σ∈Sn

sgn(σ)Aσ(1),1Aσ(2),2 · · ·Aσ(n),n,

where in the product first come the elements from the first column, then from the second and so on. The determinanton the far right is Cayley’s omega process, and the one on the left is the Capelli determinant.The operators E ᵢ can be written in a matrix form:

E = XDt,

where E,X,D are matrices with elements E ᵢ , xᵢ , ∂∂xij

respectively. If all elements in these matrices would becommutative then clearly det(E) = det(X) det(Dt) . The Capelli identity shows that despite noncommutativitythere exists a “quantization” of the formula above. The only price for the noncommutivity is a small correction:(n− i)δij on the left hand side. For generic noncommutative matrices formulas like

4

4.2. RELATIONS WITH REPRESENTATION THEORY 5

det(AB) = det(A) det(B)

do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices.That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof doesnot seem to exist. Direct verification of the statement can be given as an exercise for n' = 2, but is already long for n= 3.

4.2 Relations with representation theory

Consider the following slightly more general context. Suppose that n and m are two integers and xij for i =1, . . . , n, j = 1, . . . ,m , be commuting variables. Redefine Eij by almost the same formula:

Eij =

m∑a=1

xia∂

∂xja.

with the only difference that summation index a ranges from 1 tom . One can easily see that such operators satisfythe commutation relations:

[Eij , Ekl] = δjkEil − δilEkj .

Here [a, b] denotes the commutator ab − ba . These are the same commutation relations which are satisfied by thematrices eij which have zeros everywhere except the position (i, j) , where 1 stands. ( eij are sometimes calledmatrix units). Hence we conclude that the correspondence π : eij 7→ Eij defines a representation of the Lie algebragln in the vector space of polynomials of xij .

4.2.1 Case m = 1 and representation Sk Cn

It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:

Eij = xi∂

∂xj.

In particular, for the polynomials of the first degree it is seen that:

Eijxk = δjkxi.

Hence the action of Eij restricted to the space of first-order polynomials is exactly the same as the action of matrixunits eij on vectors in Cn . So, from the representation theory point of view, the subspace of polynomials of firstdegree is a subrepresentation of the Lie algebra gln , which we identified with the standard representation in Cn .Going further, it is seen that the differential operators Eij preserve the degree of the polynomials, and hence thepolynomials of each fixed degree form a subrepresentation of the Lie algebra gln . One can see further that the spaceof homogeneous polynomials of degree k can be identified with the symmetric tensor power SkCn of the standardrepresentation Cn .One can also easily identify the highest weight structure of these representations. The monomial xk1 is a highest weightvector, indeed: Eijxk1 = 0 for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: Eiixk1 = kδi1x

k1 .

Such representation is sometimes called bosonic representation of gln . Similar formulas Eij = ψi∂∂ψj

define theso-called fermionic representation, here ψi are anti-commuting variables. Again polynomials of k-th degree form anirreducible subrepresentation which is isomorphic to ΛkCn i.e. anti-symmetric tensor power of Cn . Highest weightof such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representationsof gln .

6 CHAPTER 4. CAPELLI’S IDENTITY

Capelli identity for m = 1

Let us return to the Capelli identity. One can prove the following:

det(E + (n− i)δij) = 0, n > 1

the motivation for this equality is the following: consider Ecij = xipj for some commuting variables xi, pj . Thematrix Ec is of rank one and hence its determinant is equal to zero. Elements of matrix E are defined by thesimilar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity:det(Ec) = 0 can be preserved for the small price of correcting matrix E by (n− i)δij .Let us also mention that similar identity can be given for the characteristic polynomial:

det(t+ E + (n− i)δij) = t[n] + Tr(E)t[n−1],

where t[k] = t(t+1) · · · (t+k−1) . The commutative counterpart of this is a simple fact that for rank = 1 matricesthe characteristic polynomial contains only the first and the second coefficients.Let us consider an example for n = 2.

∣∣∣∣t+ E11 + 1 E12

E21 t+ E22

∣∣∣∣ = ∣∣∣∣t+ x1∂1 + 1 x1∂2x2∂1 t+ x2∂2

∣∣∣∣= (t+ x1∂1 + 1)(t+ x2∂2)− x2∂1x1∂2

= t(t+ 1) + t(x1∂1 + x2∂2) + x1∂1x2∂2 + x2∂2 − x2∂1x1∂2

Using

∂1x1 = x1∂1 + 1, ∂1x2 = x2∂1, x1x2 = x2x1

we see that this is equal to:

t(t+ 1) + t(x1∂1 + x2∂2) + x2x1∂1∂2 + x2∂2 − x2x1∂1∂2 − x2∂2

= t(t+ 1) + t(x1∂1 + x2∂2) = t[2] + tTr(E).

4.2.2 The universal enveloping algebra U(gln) and its center

An interesting property of the Capelli determinant is that it commutes with all operators Eij, that is the commutator[Eij , det(E + (n− i)δij)] = 0 is equal to zero. It can be generalized:Consider any elements Eij in any ring, such that they satisfy the commutation relation [Eij , Ekl] = δjkEil − δilEkj, (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows:

det(t+ E + (n− i)δij) = t[n] +∑

k=n−1,...,0

t[k]Ck,

where t[k] = t(t+ 1) · · · (t+ k − 1),

then:

• elements Ck commute with all elements Eij

• elements Ck can be given by the formulas similar to the commutative case:

4.2. RELATIONS WITH REPRESENTATION THEORY 7

Ck =∑

I=(i1<i2<···<ik)

det(E + (k − i)δij)II ,

i.e. they are sums of principal minors of the matrix E, modulo the Capelli correction +(k − i)δij . In particularelement C0 is the Capelli determinant considered above.These statements are interrelated with the Capelli identity, as will be discussed below, and similarly to it the directfew lines short proof does not seem to exist, despite the simplicity of the formulation.The universal enveloping algebra

U(gln)

can defined as an algebra generated by

Eij

subject to the relations

[Eij , Ekl] = δjkEil − δilEkj

alone. The proposition above shows that elements Ckbelong to the center of U(gln) . It can be shown that theyactually are free generators of the center of U(gln) . They are sometimes called Capelli generators. The Capelliidentities for them will be discussed below.Consider an example for n = 2.

∣∣∣∣t+ E11 + 1 E12

E21 t+ E22

∣∣∣∣ = (t+ E11 + 1)(t+ E22)− E21E12

= t(t+ 1) + t(E11 + E22) + E11E22 − E21E12 + E22.

It is immediate to check that element (E11 + E22) commute with Eij . (It corresponds to an obvious fact that theidentity matrix commute with all other matrices). More instructive is to check commutativity of the second elementwith Eij . Let us do it for E12 :

[E12, E11E22 − E21E12 + E22]

= [E12, E11]E22 + E11[E12, E22]− [E12, E21]E12 − E21[E12, E12] + [E12, E22]

= −E12E22 + E11E12 − (E11 − E22)E12 − 0 + E12

= −E12E22 + E22E12 + E12 = −E12 + E12 = 0.

We see that the naive determinant E11E22 −E21E12 will not commute with E12 and the Capelli’s correction+E22

is essential to ensure the centrality.

4.2.3 General m and dual pairs

Let us return to the general case:

Eij =m∑a=1

xia∂

∂xja,

for arbitrary n and m. Definition of operators E ᵢ can be written in a matrix form: E = XDt , where E is n × nmatrix with elements Eij ; X is n×m matrix with elements xij ; D is n×m matrix with elements ∂

∂xij.

8 CHAPTER 4. CAPELLI’S IDENTITY

Capelli–Cauchy–Binet identitiesFor general m matrix E is given as product of the two rectangular matrices: X and transpose to D. If all elements ofthese matrices would commute then one knows that the determinant of E can be expressed by the so-called Cauchy–Binet formula via minors of X and D. An analogue of this formula also exists for matrix E again for the same mildprice of the correction E → (E + (n− i)δij) :

det(E + (n− i)δij) =∑

I=(1≤i1<i2<···<in≤m)

det(XI) det(DtI)

In particular (similar to the commutative case): if m<n'', then <math>\det(E+(n-i)\delta_ij) =0 </math>; if m=nwe return to the identity above.

Let us also mention that similar to the commutative case (see Cauchy–Binet for minors), one can express not onlythe determinant of E, but also its minors via minors of X and D:

det(E + (s− i)δij)KL =∑

I=(1≤i1<i2<···<is≤m)

det(XKI) det(DtIL)

Here K = (k1 < k2 < ... < ks), L = (l1 < l2 < ... < ls), are arbitrary multi-indexes; as usuallyMKL denotes a submatrixof M formed by the elements Mkalb. Pay attention that the Capelli correction now contains s, not n as in previousformula. Note that for s=1, the correction (s − i) disappears and we get just the definition of E as a product of X andtranspose to D. Let us also mention that for generic K,L corresponding minors do not commute with all elements Eij,so the Capelli identity exists not only for central elements.As a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention thefollowing:

det(t+ E + (n− i)δij) = t[n] +∑

k=n−1,...,0

t[k]∑I,J

det(XIJ) det(DtJI),

where I = (1 ≤ i1 < · · · < ik ≤ n), J = (1 ≤ j1 < · · · < jk ≤ n) . This formula is similar to the commutativecase, modula +(n− i)δij at the left hand side and t[n] instead of tn at the right hand side.Relation to dual pairsModern interest in these identities has been much stimulated by Roger Howe who considered them in his theoryof reductive dual pairs (also known as Howe duality). To make the first contact with these ideas, let us look moreprecisely on operators Eij . Such operators preserve the degree of polynomials. Let us look at the polynomials ofdegree 1: Eijxkl = xilδjk , we see that index l is preserved. One can see that from the representation theory pointof view polynomials of the first degree can be identified with direct sum of the representations Cn ⊕ · · · ⊕Cn , herel-th subspace (l=1...m) is spanned by xil , i = 1, ..., n. Let us give another look on this vector space:

Cn ⊕ · · · ⊕ Cn = Cn ⊗ Cm.

Such point of view gives the first hint of symmetry between m and n. To deepen this idea let us consider:

Edualij =

n∑a=1

xai∂

∂xaj.

These operators are given by the same formulas asEij modula renumeration i↔ j , hence by the same arguments wecan deduce that Edual

ij form a representation of the Lie algebra glm in the vector space of polynomials of xij. Beforegoing further we can mention the following property: differential operatorsEdual

ij commute with differential operatorsEkl .The Lie groupGLn×GLm acts on the vector spaceCn⊗Cm in a natural way. One can show that the correspondingaction of Lie algebra gln × glm is given by the differential operators Eij and Edual

ij respectively. This explains thecommutativity of these operators.The following deeper properties actually hold true:

4.3. GENERALIZATIONS 9

• The only differential operators which commute with Eij are polynomials in Edualij , and vice versa.

• Decomposition of the vector space of polynomials into a direct sum of tensor products of irreducible repre-sentations of GLn and GLm can be given as follows:

C[xij ] = S(Cn ⊗ Cm) =∑D

ρDn ⊗ ρD′

m .

The summands are indexed by the Young diagrams D, and representations ρD are mutually non-isomorphic. Anddiagram D determine D′ and vice versa.

• In particular the representation of the big group GLn × GLm is multiplicity free, that is each irreduciblerepresentation occurs only one time.

One easily observe the strong similarity to Schur–Weyl duality.

4.3 Generalizations

Much work have been done on the identity and its generalizations. Approximately two dozens of mathematicians andphysicists contributed to the subject, to name a few: R. Howe, B. Kostant[1][2] Fields medalist A. Okounkov[3][4] A.Sokal,[5] D. Zeilberger.[6]

It seems historically the first generalizations were obtained by Herbert Westren Turnbull in 1948,[7] who found thegeneralization for the case of symmetric matrices (see[5][6] for modern treatments).The other generalizations can be divided into several patterns. Most of them are based on the Lie algebra point of view.Such generalizations consist of changing Lie algebra gln to simple Lie algebras [8] and their super[9][10] (q),[11][12] andcurrent versions.[13] As well as identity can be generalized for different reductive dual pairs.[14][15] And finally one canconsider not only the determinant of the matrix E, but its permanent,[16] trace of its powers and immanants.[3][4][17][18]Let us mention few more papers;[19][20][21] [22] [23] [24] [25] still the list of references is incomplete. It has been believedfor quite a long time that the identity is intimately related with semi-simple Lie algebras. Surprisingly a new purelyalgebraic generalization of the identity have been found in 2008[5] by S. Caracciolo, A. Sportiello, A. D. Sokal whichhas nothing to do with any Lie algebras.

4.3.1 Turnbull’s identity for symmetric matrices

Consider symmetric matrices

X =

∣∣∣∣∣∣∣∣∣∣∣

x11 x12 x13 · · · x1nx12 x22 x23 · · · x2nx13 x23 x33 · · · x3n...

...... . . . ...

x1n x2n x3n · · · xnn

∣∣∣∣∣∣∣∣∣∣∣, D =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

2 ∂∂x11

∂∂x12

∂∂x13

· · · ∂∂x1n

∂∂x12

2 ∂∂x22

∂∂x23

· · · ∂∂x2n

∂∂x13

∂∂x23

2 ∂∂x33

· · · ∂∂x3n

......

... . . . ...∂

∂x1n

∂∂x2n

∂∂x3n

· · · 2 ∂∂xnn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣Herbert Westren Turnbull[7] in 1948 discovered the following identity:

det(XD + (n− i)δij) = det(X) det(D)

Combinatorial proof can be found in the paper,[6] another proof and amusing generalizations in the paper,[5] see alsodiscussion below.

10 CHAPTER 4. CAPELLI’S IDENTITY

4.3.2 The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices

Consider antisymmetric matrices

X =

∣∣∣∣∣∣∣∣∣∣∣

0 x12 x13 · · · x1n−x12 0 x23 · · · x2n−x13 −x23 0 · · · x3n...

...... . . . ...

−x1n −x2n −x3n · · · 0

∣∣∣∣∣∣∣∣∣∣∣, D =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 ∂∂x12

∂∂x13

· · · ∂∂x1n

− ∂∂x12

0 ∂∂x23

· · · ∂∂x2n

− ∂∂x13

− ∂∂x23

0 · · · ∂∂x3n

......

... . . . ...− ∂∂x1n

− ∂∂x2n

− ∂∂x3n

· · · 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

Then

det(XD + (n− i)δij) = det(X) det(D).

4.3.3 The Caracciolo–Sportiello–Sokal identity for Manin matrices

Consider two matrices M and Y over some associative ring which satisfy the following condition

[Mij , Ykl] = −δjkQil

for some elements Qil. Or ”in words”: elements in j-th column ofM commute with elements in k-th row of Y unlessj = k, and in this case commutator of the elements Mik and Ykl depends only on i, l, but does not depend on k.Assume that M is a Manin matrix (the simplest example is the matrix with commuting elements).Then for the square matrix case

det(MY +Q diag(n− 1, n− 2, . . . , 1, 0)) = det(M) det(Y ).

Here Q is a matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n− 1, n − 2, ..., 1, 0 on the diagonal.See [5] proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.Obviously the original Cappeli’s identity the particular case of this identity. Moreover from this identity one can seethat in the original Capelli’s identity one can consider elements

∂

∂xij+ fij(x11, . . . , xkl, . . . )

for arbitrary functions fij and the identity still will be true.

4.3.4 The Mukhin–Tarasov–Varchenko identity and the Gaudin model

Statement

Consider matrices X and D as in Capelli’s identity, i.e. with elements xij and ∂ij at position (ij).Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complexnumbers.

det(∂

∂z−A−X

1

z −BDt

)

4.3. GENERALIZATIONS 11

= detcommute all if as calculateall Putx and zright the on derivations all while left, the on(∂

∂z−A−X

1

z −BDt

)Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries.The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, whilederivations on the right. (Such recipe is called a Wick ordering in the quantum mechanics).

The Gaudin quantum integrable system and Talalaev’s theorem

The matrix

L(z) = A+X1

z −BDt

is a Lax matrix for the Gaudin quantum integrable spin chain system. D. Talalaev solved the long-standing problemof the explicit solution for the full set of the quantum commuting conservation laws for the Gaudin model, discoveringthe following theorem.Consider

det(∂

∂z− L(z)

)=

n∑i=0

Hi(z)

(∂

∂z

)i.

Then for all i,j,z,w

[Hi(z),Hj(w)] = 0,

i.e. Hi(z) are generating functions in z for the differential operators in x which all commute. So they provide quantumcommuting conservation laws for the Gaudin model.

4.3.5 Permanents, immanants, traces – “higher Capelli identities”

The original Capelli identity is a statement about determinants. Later, analogous identities were found for permanents,immanants and traces. Based on the combinatorial approach paper by S.G. Williamson [26] was one of the first resultsin this direction.

Turnbull’s identity for permanents of antisymmetric matrices

Consider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of theHUKS identity above.Then

perm(XtD − (n− i)δij) = permcommute all if as Calculateall Putxright the on derivations all with left, the on (X

tD).

Let us cite:[6] "...is stated without proof at the end of Turnbull’s paper”. The authors themselves follow Turnbull – atthe very end of their paper they write:“Since the proof of this last identity is very similar to the proof of Turnbull’s symmetric analog (with a slight twist),we leave it as an instructive and pleasant exercise for the reader.”.The identity is deeply analyzed in paper .[27]

12 CHAPTER 4. CAPELLI’S IDENTITY

4.4 References[1] Kostant, B.; Sahi, S. (1991), “The Capelli Identity, tube domains, and the generalized Laplace transform”, Advances in

Math. 87: 71–92, doi:10.1016/0001-8708(91)90062-C

[2] Kostant, B.; Sahi, S. (1993), “Jordan algebras andCapelli identities”, InventionesMathematicae 112 (1): 71–92, doi:10.1007/BF01232451

[3] Okounkov, A. (1996), Quantum Immanants and Higher Capelli Identities, arXiv:q-alg/9602028

[4] Okounkov, A. (1996), Young Basis, Wick Formula, and Higher Capelli Identities, arXiv:q-alg/9602027

[5] Caracciolo, S.; Sportiello, A.; Sokal, A. (2008), Noncommutative determinants, Cauchy–Binet formulae, and Capelli-typeidentities. I. Generalizations of the Capelli and Turnbull identities, arXiv:0809.3516

[6] Foata, D.; Zeilberger, D. (1993), Combinatorial Proofs of Capelli’s and Turnbull’s Identities from Classical Invariant Theory,arXiv:math/9309212

[7] Turnbull, HerbertWestren (1948), “Symmetric determinants and the Cayley and Capelli operators”, Proc. EdinburghMath.Soc. 8 (2): 76–86, doi:10.1017/S0013091500024822

[8] Molev, A.; Nazarov, M. (1997), Capelli Identities for Classical Lie Algebras, arXiv:q-alg/9712021

[9] Molev, A. (1996), Factorial supersymmetric Schur functions and super Capelli identities, arXiv:q-alg/9606008

[10] Nazarov, M. (1996), Capelli identities for Lie superalgebras, arXiv:q-alg/9610032

[11] Noumi, M.; Umeda, T.; Wakayma, M. (1994), “A quantum analogue of the Capelli identity and an elementary differentialcalculus on GLq(n)", Duke Mathematical Journal 76 (2): 567–594, doi:10.1215/S0012-7094-94-07620-5

[12] Noumi, M.; Umeda, T.; Wakayma, M. (1996), “Dual pairs, spherical harmonics and a Capelli identity in quantum grouptheory”, Compositio Mathematica 104 (2): 227–277

[13] Mukhin, E.; Tarasov, V.; Varchenko, A. (2006), A generalization of the Capelli identity, arXiv:math.QA/0610799

[14] Itoh,M. (2004), “Capelli identities for reductive dual pairs”,Advances inMathematics 194 (2): 345–397, doi:10.1016/j.aim.2004.06.010

[15] Itoh, M. (2005), “Capelli Identities for the dual pair ( O M, Sp N)", Mathematische Zeitschrift 246 (1–2): 125–154,doi:10.1007/s00209-003-0591-2

[16] Nazarov, M. (1991), “Quantum Berezinian and the classical Capelli identity”, Letters in Mathematical Physics 21 (2): 123–131, doi:10.1007/BF00401646

[17] Nazarov, M. (1996), Yangians and Capelli identities, arXiv:q-alg/9601027

[18] Molev, A. (1996), A Remark on the Higher Capelli Identities, arXiv:q-alg/9603007

[19] Kinoshita, K.; Wakayama, M. (2002), “Explicit Capelli identities for skew symmetric matrices”, Proceedings of the Edin-burgh Mathematical Society 45 (2): 449–465, doi:10.1017/S0013091500001176

[20] Hashimoto, T. (2008),Generating function for GLn-invariant differential operators in the skewCapelli identity, arXiv:0803.1339

[21] Nishiyama, K.; Wachi, A. (2008), A note on the Capelli identities for symmetric pairs of Hermitian type, arXiv:0808.0607

[22] Umeda, Toru (2008), “On the proof of the Capelli identities”, Funkcialaj Ekvacioj 51 (1): 1–15, doi:10.1619/fesi.51.1

[23] Brini, A; Teolis, A (1993), “Capelli’s theory, Koszulmaps, and superalgebras”, PNAS 90 (21): 10245–10249, doi:10.1073/pnas.90.21.10245

[24] Koszul, J (1981), “Les algebres de Lie graduées de type sl (n, 1) et l'opérateur de A. Capelli”, C.R. Acad. Sci. Paris (292):139–141

[25] Orsted, B; Zhang, G (2001), Capelli identity and relative discrete series of line bundles over tube domains

[26] Williamson, S. (1981), “Symmetry operators, polarizations, and a generalized Capelli identity”, Linear & Multilinear Al-gebra 10 (2): 93–102, doi:10.1080/03081088108817399

[27] Umeda, Toru (2000), “On Turnbull identity for skew-symmetric matrices”, Proc. Edinburgh Math. Soc. 43 (2): 379–393,doi:10.1017/S0013091500020988

4.5. FURTHER READING 13

4.5 Further reading• Capelli, Alfredo (1887), “Ueber die Zurückführung der Cayley’schen Operation Ω auf gewöhnliche Polar-Operationen”,Mathematische Annalen (Berlin / Heidelberg: Springer) 29 (3): 331–338, doi:10.1007/BF01447728,ISSN 1432-1807

• Howe, Roger (1989), “Remarks on classical invariant theory”, Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539–570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR2001418, MR 0986027

• Howe, Roger; Umeda, Toru (1991), “The Capelli identity, the double commutant theorem, and multiplicity-free actions”, Mathematische Annalen 290 (1): 565–619, doi:10.1007/BF01459261

• Umeda, Tôru (1998), “The Capelli identities, a century after”, Selected papers on harmonic analysis, groups,and invariants, Amer. Math. Soc. Transl. Ser. 2 183, Providence, R.I.: Amer. Math. Soc., pp. 51–78, ISBN978-0-8218-0840-5, MR 1615137

• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255, retrieved 03/2007/26 Check date values in: |accessdate=(help)

Chapter 5

Catalecticant

But the catalecticant of the biquadratic function of x, y was first brought into notice as an invariant by Mr Boole;and the discriminant of the quadratic function of x, y is identical with its catalecticant, as also with its Hessian.Meicatalecticizant would more completely express the meaning of that which, for the sake of brevity, I denominatethe catalecticant.Sylvester (1852), quoted by (Miller 2010)

In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coefficients thatvanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced bySylvester (1852); see (Miller 2010). The word catalectic refers to an incomplete line of verse, lacking a syllable atthe end or ending with an incomplete foot.

5.1 Binary forms

The catalecticant of a binary form of degree 2n is a polynomial in its coefficients that vanishes when the binary formis a sum of at most n powers of linear forms Sturmfels (1993).The catalecticant of a binary form can be given as the determinant of a catalecticant matrix (Eisenbud 1988), alsocalled a Hankel matrix, that is a square matrix with constant (positive sloping) skew-diagonals, such as

a b c d eb c d e fc d e f gd e f g he f g h i

.

5.2 Catalecticants of quartic forms

The catalecticant of a quartic form is the resultant of its second partial derivatives. For binary quartics the catalecticantvanishes when the form is a sum of 2 4th powers. For a ternary quartic the catalecticant vanishes when the form is asum of 5 4th powers. For quaternary quartics the catalecticant vanishes when the form is a sum of 9 4th powers. Forquinary quartics the catalecticant vanishes when the form is a sum of 14 4th powers. (Elliot 1915, p.295)

5.3 References

• Eisenbud, David (1988), “Linear sections of determinantal varieties”, American Journal of Mathematics 110(3): 541–575, doi:10.2307/2374622, ISSN 0002-9327, MR 944327

14

5.4. EXTERNAL LINKS 15

• Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. (2nd ed.), Oxford. ClarendonPress, JFM 26.0135.01

• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN978-3-211-82445-0,MR1255980

• Sylvester, J. J. (1852), “On the principles of the calculus of forms”, Cambridge and Dublin MathematicalJournal: 52–97

5.4 External links• Miller, Jeff (2010), Earliest Known Uses of Some of the Words of Mathematics (C)

Chapter 6

Cayley’s Ω process

This article is about the mathematical process. For the industrial OMEGA process, see OMEGA process.

Inmathematics,Cayley’sΩ process, introduced byArthur Cayley (1846), is a relatively invariant differential operatoron the general linear group, that is used to construct invariants of a group action.As a partial differential operator acting on functions of n2 variables xij, the omega operator is given by the determinant

Ω =

∣∣∣∣∣∣∣∂

∂x11· · · ∂

∂x1n... . . . ...∂

∂xn1· · · ∂

∂xnn

∣∣∣∣∣∣∣.For binary forms f in x1, y1 and g in x2, y2 the Ω operator is ∂2fg

∂x1∂y2− ∂2fg

∂x2∂y1. The r-fold Ω process Ωr(f, g) on

two forms f and g in the variables x and y is then

1. Convert f to a form in x1, y1 and g to a form in x2, y2

2. Apply the Ω operator r times to the function fg, that is, f times g in these four variables

3. Substitute x for x1 and x2, y for y1 and y2 in the result

The result of the r-fold Ω process Ωr(f, g) on the two forms f and g is also called the r-th transvectant and is commonlywritten (f, g)r.

6.1 Applications

Cayley’s Ω process appears in Capelli’s identity, which Weyl (1946) used to find generators for the invariants ofvarious classical groups acting on natural polynomial algebras.Hilbert (1890) used Cayley’s Ω process in his proof of finite generation of rings of invariants of the general lineargroup. His use of the Ω process gives an explicit formula for the Reynolds operator of the special linear group.Cayley’s Ω process is used to define transvectants.

6.2 References

• Cayley, Arthur (1846), “On linear transformations”, Cambridge and Dublin mathematical journal 1: 104–122Reprinted in Cayley (1889), The collected mathematical papers 1, Cambridge: Cambridge University press,pp. 95–112

16

6.2. REFERENCES 17

• Hilbert, David (1890), “Ueber die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831

• Howe, Roger (1989), “Remarks on classical invariant theory.”, Transactions of the American Mathematical So-ciety (American Mathematical Society) 313 (2): 539–570, doi:10.1090/S0002-9947-1989-0986027-X, ISSN0002-9947, JSTOR 2001418, MR 0986027

• Olver, Peter J. (1999), Classical invariant theory, Cambridge University Press, ISBN 978-0-521-55821-1

• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, New York: Springer-Verlag, ISBN 978-3-211-82445-0, MR 1255980

• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255, retrieved 03/2007/26 Check date values in: |accessdate=(help)

Chapter 7

Chevalley–Iwahori–Nagata theorem

In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearlyon a finite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomialsis an isomorphism if this ring is finitely generated and all orbits of G on V are closed (Dieudonné & Carrell 1970,1971, p.55). It is named after Claude Chevalley, Nagayoshi Iwahori, and Masayoshi Nagata.

7.1 References• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525

• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

18

Chapter 8

Chevalley–Shephard–Todd theorem

In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring ofinvariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generatedby pseudoreflections. In the case of subgroups of the complex general linear group the theorem was first proved byG. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwardsgave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case byJean-Pierre Serre.

8.1 Statement of the theorem

Let V be a finite-dimensional vector space over a field K and let G be a finite subgroup of the general linear groupGL(V). An element s of GL(V) is called a pseudoreflection if it fixes a codimension 1 subspace of V and is not theidentity transformation I, or equivalently, if the kernel Ker (s − I) has codimension one in V. Assume that the orderof G is relatively prime to the characteristic of K (the so-called non-modular case). Then the following properties areequivalent:[1]

• (A) The group G is generated by pseudoreflections.

• (B) The algebra of invariants K[V]G is a (free) polynomial algebra.

• (B′) The algebra of invariants K[V]G is a regular ring.

• (C) The algebra K[V] is a free module over K[V]G.

• (C′) The algebra K[V] is a projective module over K[V]G.

In the case when the field K is the field C of complex numbers, the first condition is usually stated as "G is a complexreflection group". Shephard and Todd derived a full classification of such groups.

8.2 Examples• Let V be one-dimensional. Then any finite group faithfully acting on V is a subgroup of the multiplicativegroup of the field K, and hence a cyclic group. It follows that G consists of roots of unity of order dividing n,where n is its order, so G is generated by pseudoreflections. In this case, K[V] = K[x] is the polynomial ringin one variable and the algebra of invariants of G is the subalgebra generated by xn, hence it is a polynomialalgebra.

• LetV =Kn be the standard n-dimensional vector space andG be the symmetric group Sn acting by permutationsof the elements of the standard basis. The symmetric group is generated by transpositions (ij), which act byreflections on V. On the other hand, by the main theorem of symmetric functions, the algebra of invariants isthe polynomial algebra generated by the elementary symmetric functions e1, … en.

19

20 CHAPTER 8. CHEVALLEY–SHEPHARD–TODD THEOREM

• Let V = K2 and G be the cyclic group of order 2 acting by ±I. In this case, G is not generated by pseudoreflec-tions, since the nonidentity element s of G acts without fixed points, so that dim Ker (s − I) = 0. On the otherhand, the algebra of invariants is the subalgebra of K[V] = K[x, y] generated by the homogeneous elementsx2, xy, and y2 of degree 2. This subalgebra is not a polynomial algebra because of the relation x2y2 = (xy)2.

8.3 Generalizations

Broer (2007) gave an extension of the Chevalley–Shephard–Todd theorem to positive characteristic.There has been much work on the question of when a reductive algebraic group acting on a vector space has apolynomial ring of invariants. In the case when the algebraic group is simple all cases when the invariant ring ispolynomial have been classified by Schwarz (1978)In general, the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so itis a finite rank free module over a polynomial subring.

8.4 Notes[1] See, e.g.: Bourbaki, Lie, chap. V, §5, nº5, theorem 4 for equivalence of (A), (B) and (C); page 26 of for equivalence of

(A) and (B′); pages 6–18 of for equivalence of (C) and (C′) for a proof of (B′)⇒(A).

8.5 References• Bourbaki, Nicolas, Éléments de mathématiques : Groupes et algèbres de Lie (English translation: Bourbaki,Nicolas, Elements of Mathematics: Lie Groups and Lie Algebras)

• Broer, Abraham (2007), On Chevalley-Shephard-Todd’s theorem in positive characteristic, [], arXiv:0709.0715

• Chevalley, Claude (1955), “Invariants of finite groups generated by reflections”, Amer. J. Of Math. 77 (4):778–782, doi:10.2307/2372597, JSTOR 2372597

• Neusel, Mara D.; Smith, Larry (2002), Invariant Theory of Finite Groups, American Mathematical Society,ISBN 0-8218-2916-5

• Shephard, G. C.; Todd, J. A. (1954), “Finite unitary reflection groups”, Canadian J. Math. 6: 274–304,doi:10.4153/CJM-1954-028-3

• Schwarz, G. (1978), “Representations of simple Lie groups with regular rings of invariants”, Invent. Math. 49(2): 167–191, doi:10.1007/BF01403085

• Smith, Larry (1997), “Polynomial invariants of finite groups. A survey of recent developments”, Bull. Amer.Math. Soc. 34 (3): 211–250, doi:10.1090/S0273-0979-97-00724-6, MR 1433171

• Springer, T. A. (1977), Invariant Theory, Springer, ISBN 0-387-08242-5

Chapter 9

Differential invariant

In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves thederivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential ge-ometry, and the curvature is often studied from this point of view.[1] Differential invariants were introduced in specialcases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. Lie (1884) was thefirst general work on differential invariants, and established the relationship between differential invariants, invariantdifferential equations, and invariant differential operators.Differential invariants are contrasted with geometric invariants. Whereas differential invariants can involve a distin-guished choice of independent variables (or a parameterization), geometric invariants do not. Élie Cartan's methodof moving frames is a refinement that, while less general than Lie’s methods of differential invariants, always yieldsinvariants of the geometrical kind.

9.1 Definition

The simplest case is for differential invariants for one independent variable x and one dependent variable y. Let Gbe a Lie group acting on R2. Then G also acts, locally, on the space of all graphs of the form y = ƒ(x). Roughlyspeaking, a k-th order differential invariant is a function

I

(x, y,

dy

dx, . . . ,

dky

dxk

)depending on y and its first k derivatives with respect to x, that is invariant under the action of the group.The group can act on the higher-order derivatives in a nontrivial manner that requires computing the prolongation ofthe group action. The action of G on the first derivative, for instance, is such that the chain rule continues to hold: if

(x, y) = g · (x, y),

then

g ·(x, y,

dy

dx

)def=

(x, y,

dy

dx

).

Similar considerations apply for the computation of higher prolongations. This method of computing the prolongationis impractical, however, and it is much simpler to work infinitesimally at the level of Lie algebras and the Lie derivativealong the G action.More generally, differential invariants can be considered for mappings from any smooth manifold X into anothersmooth manifold Y for a Lie group acting on the Cartesian product X×Y. The graph of a mapping X → Y is asubmanifold of X×Y that is everywhere transverse to the fibers over X. The group G acts, locally, on the space of

21

22 CHAPTER 9. DIFFERENTIAL INVARIANT

such graphs, and induces an action on the k-th prolongation Y (k) consisting of graphs passing through each pointmodulo the relation of k-th order contact. A differential invariant is a function on Y (k) that is invariant under theprolongation of the group action.

9.2 Applications• Differential invariants can be applied to the study of systems of partial differential equations: seeking similaritysolutions that are invariant under the action of a particular group can reduce the dimension of the problem (i.e.yield a “reduced system”).[2]

• Noether’s theorem implies the existence of differential invariants corresponding to every differentiable sym-metry of a variational problem.

• Flow characteristics using computer vision[3]

• Geometric integration

9.3 See also• Cartan’s equivalence method

9.4 Notes[1] Guggenheimer 1977

[2] Olver 1994, Chapter 3

[3] http://dspace.mit.edu/bitstream/handle/1721.1/3348/P-2219-29812804.pdf?sequence=1

9.5 References• Guggenheimer, Heinrich (1977), Differential Geometry, New York: Dover Publications, ISBN 978-0-486-63433-3.

• Lie, Sophus (1884), "Über Differentialinvarianten”, Gesammelte Adhandlungen 6, Leipzig: B.G. Teubner,pp. 95–138; English translation: Ackerman, M; Hermann, R (1975), Sophus Lie’s 1884 Differential InvariantPaper, Brookline, Mass.: Math Sci Press.

• Olver, Peter J. (1993),Applications of Lie groups to differential equations (2nd ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-94007-6.

• Olver, Peter J. (1995), Equivalents, Invariants, and Symmetry, Cambridge University Press, ISBN 978-0-521-47811-3.

• Mansfield, Elizabeth Louise (2009), A Practical Guide to the Invariant Calculus (PDF); to be published byCambridge 2010, ISBN 978-0-521-85701-7.

9.6 External links• Invariant Variation Problems

Chapter 10

Evectant

In mathematical invariant theory, an evectant is a contravariant constructed from an invariant by acting on it with adifferential operator called an evector. Evectants and evectors were introduced by Sylvester (1854, p.95).

10.1 References• Sylvester, James Joseph (1853), “On the calculus of forms, otherwise the theory of invariants”, The Cambridgeand Dublin Mathematical Journal 8: 257–269

• Sylvester, James Joseph (1854), “On the calculus of forms, otherwise the theory of invariants”, The Cambridgeand Dublin Mathematical Journal 9: 85–103

23

Chapter 11

Geometric invariant theory

In mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions inalgebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas fromthe paper (Hilbert 1893) in classical invariant theory.Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides tech-niques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to constructmoduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980sthe theory developed interactions with symplectic geometry and equivariant topology, and was used to constructmoduli spaces of objects in differential geometry, such as instantons and monopoles.

11.1 Background

Main article: Invariant theory

Invariant theory is concerned with a group action of a group G on an algebraic variety (or a scheme) X. Classicalinvariant theory addresses the situation when X = V is a vector space and G is either a finite group, or one of theclassical Lie groups that acts linearly on V. This action induces a linear action of G on the space of polynomialfunctions R(V) on V by the formula

g · f(v) = f(g−1v), g ∈ G, v ∈ V.

The polynomial invariants of the G-action on V are those polynomial functions f on V which are fixed under the'change of variables’ due to the action of the group, so that g·f = f for all g in G. They form a commutative algebraA = R(V)G, and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' V //G. In thelanguage of modern algebraic geometry,

V //G = SpecA = SpecR(V )G.

Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a generallinear group, is to prove that the algebra A is finitely generated. This is necessary if one wanted the quotient to be anaffine algebraic variety. Whether a similar fact holds for arbitrary groups G was the subject of Hilbert’s fourteenthproblem, and Nagata demonstrated that the answer was negative in general. On the other hand, in the course ofdevelopment of representation theory in the first half of the twentieth century, a large class of groups for which theanswer is positive was identified; these are called reductive groups and include all finite groups and all classical groups.The finite generation of the algebra A is but the first step towards the complete description of A, and the progressin resolving this more delicate question was rather modest. The invariants had classically been described only in arestricted range of situations, and the complexity of this description beyond the first few cases held out little hope forfull understanding of the algebras of invariants in general. Furthermore, it may happen that all polynomial invariantsf take the same value on a given pair of points u and v in V, yet these points are in different orbits of the G-action. A

24

11.2. MUMFORD’S BOOK 25

simple example is provided by themultiplicative groupC* of non-zero complex numbers that acts on an n-dimensionalcomplex vector space Cn by scalar multiplication. In this case, every polynomial invariant is a constant, but there aremany different orbits of the action. The zero vector forms an orbit by itself, and the non-zeromultiples of any non-zerovector form an orbit, so that non-zero orbits are paramatrized by the points of the complex projective spaceCPn−1. Ifthis happens, one says that “invariants do not separate the orbits”, and the algebra A reflects the topological quotientspace X /G rather imperfectly. Indeed, the latter space is frequently non-separated. In 1893 Hilbert formulated andproved a criterion for determining those orbits which are not separated from the zero orbit by invariant polynomials.Rather remarkably, unlike his earlier work in invariant theory, which led to the rapid development of abstract algebra,this result of Hilbert remained little known and little used for the next 70 years. Much of the development of invarianttheory in the first half of the twentieth century concerned explicit computations with invariants, and at any rate,followed the logic of algebra rather than geometry.

11.2 Mumford’s book

Geometric invariant theory was founded and developed by Mumford in a monograph, first published in 1965, thatapplied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometryquestions. (The book was greatly expanded in two later editions, with extra appendices by Fogarty and Mumford,and a chapter on symplectic quotients by Kirwan.) The book uses both scheme theory and computational techniquesavailable in examples. The abstract setting used is that of a group action on a scheme X. The simple-minded idea ofan orbit space

G\X,

i.e. the quotient space of X by the group action, runs into difficulties in algebraic geometry, for reasons that areexplicable in abstract terms. There is in fact no general reason why equivalence relations should interact well with the(rather rigid) regular functions (polynomial functions), which are at the heart of algebraic geometry. The functionson the orbit space G\X that should be considered are those on X that are invariant under the action of G. The directapproach can be made, by means of the function field of a variety (i.e. rational functions): take the G-invariantrational functions on it, as the function field of the quotient variety. Unfortunately this — the point of view ofbirational geometry — can only give a first approximation to the answer. As Mumford put it in the Preface to thebook:

The problem is, within the set of all models of the resulting birational class, there is one model whosegeometric points classify the set of orbits in some action, or the set of algebraic objects in some moduliproblem.

In Chapter 5 he isolates further the specific technical problem addressed, in a moduli problem of quite classicaltype — classify the big 'set' of all algebraic varieties subject only to being non-singular (and a requisite condition onpolarization). The moduli are supposed to describe the parameter space. For example for algebraic curves it has beenknown from the time of Riemann that there should be connected components of dimensions

0, 1, 3, 6, 9, …

according to the genus g =0, 1, 2, 3, 4, …, and the moduli are functions on each component. In the coarse moduliproblem Mumford considers the obstructions to be:

• non-separated topology on the moduli space (i.e. not enough parameters in good standing)

• infinitely many irreducible components (which isn't avoidable, but local finiteness may hold)

• failure of components to be representable as schemes, although respectable topologically.

It is the third point that motivated the whole theory. As Mumford puts it, if the first two difficulties are resolved

[the third question] becomes essentially equivalent to the question of whether an orbit space of some locallyclosed subset of the Hilbert or Chow schemes by the projective group exists.

26 CHAPTER 11. GEOMETRIC INVARIANT THEORY

To deal with this he introduced a notion (in fact three) of stability. This enabled him to open up the previouslytreacherous area — much had been written, in particular by Francesco Severi, but the methods of the literaturehad limitations. The birational point of view can afford to be careless about subsets of codimension 1. To havea moduli space as a scheme is on one side a question about characterising schemes as representable functors (asthe Grothendieck school would see it); but geometrically it is more like a compactification question, as the stabilitycriteria revealed. The restriction to non-singular varieties will not lead to a compact space in any sense as modulispace: varieties can degenerate to having singularities. On the other hand the points that would correspond to highlysingular varieties are definitely too 'bad' to include in the answer. The correct middle ground, of points stable enoughto be admitted, was isolated by Mumford’s work. The concept was not entirely new, since certain aspects of it wereto be found in David Hilbert's final ideas on invariant theory, before he moved on to other fields.The book’s Preface also enunciated the Mumford conjecture, later proved by William Haboush.

11.3 Stability

“Stable point” redirects here. It is not to be confused with Stable fixed point.

If a reductive group G acts linearly on a vector space V, then a non-zero point of V is called

• unstable if 0 is in the closure of its orbit,

• semi-stable if 0 is not in the closure of its orbit,

• stable if its orbit is closed, and its stabilizer is finite.

There are equivalent ways to state these (this criterion is known as the Hilbert–Mumford criterion):

• A non-zero point x is unstable if and only if there is a 1-parameter subgroup of G all of whose weights withrespect to x are positive.

• A non-zero point x is unstable if and only if every invariant polynomial has the same value on 0 and x.

• A non-zero point x is semistable if and only if there is no 1-parameter subgroup of G all of whose weights withrespect to x are positive.

• A non-zero point x is semistable if and only if some invariant polynomial has different values on 0 and x.

• A non-zero point x is stable if and only if every 1-parameter subgroup of G has positive (and negative) weightswith respect to x.

• A non-zero point x is stable if and only if for every y not in the orbit of x there is some invariant polynomial thathas different values on y and x, and the ring of invariant polynomials has transcendence degree dim(V)−dim(G).

A point of the corresponding projective space of V is called unstable, semi-stable, or stable if it is the image of apoint in V with the same property. “Unstable” is the opposite of “semistable” (not “stable”). The unstable pointsform a Zariski closed set of projective space, while the semistable and stable points both form Zariski open sets(possibly empty). These definitions are from (Mumford 1977) and are not equivalent to the ones in the first editionof Mumford’s book.Many moduli spaces can be constructed as the quotients of the space of stable points of some subset of projectivespace by some group action. These spaces can often compactified by adding certain equivalence classes of semistablepoints. Different stable orbits correspond to different points in the quotient, but two different semistable orbits maycorrespond to the same point in the quotient if their closures intersect.Example: (Deligne & Mumford 1969) A stable curve is a reduced connected curve of genus ≥2 such that its onlysingularities are ordinary double points and every non-singular rational component meets the other components inat least 3 points. The moduli space of stable curves of genus g is the quotient of a subset of the Hilbert scheme ofcurves in P5g−6 with Hilbert polynomial (6n−1)(g−1) by the group PGL₅g₋₅.Example: A vector bundle W over an algebraic curve (or over a Riemann surface) is a stable vector bundle if andonly if

11.4. SEE ALSO 27

deg(V )

rank(V )<

deg(W )

rank(W )

for all proper non-zero subbundles V ofW and is semistable if this condition holds with < replaced by ≤.

11.4 See also• Geometric complexity theory

• Geometric quotient

• Categorical quotient

• Quantization commutes with reduction

11.5 References• Deligne, Pierre; Mumford, David (1969), “The irreducibility of the space of curves of given genus”, PublicationsMathématiques de l'IHÉS 36 (36): 75–109, doi:10.1007/BF02684599, MR 0262240

• Hilbert, D. (1893), "Über die vollen Invariantensysteme”,Math. Annalen 42 (3): 313, doi:10.1007/BF01444162

• Kirwan, Frances, Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, 31.Princeton University Press, Princeton, NJ, 1984. i+211 pp. MR 0766741 ISBN 0-691-08370-3

• Kraft, Hanspeter, Geometrische Methoden in der Invariantentheorie. (German) (Geometrical methods in in-variant theory) Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. x+308 pp. MR0768181 ISBN 3-528-08525-8

• Mumford, David (1977), “Stability of projective varieties”, L'Enseignement Mathématique. Revue Interna-tionale. IIe Série 23 (1): 39–110, ISSN 0013-8584, MR 0450272

• Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematikund ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)] 34 (3rd ed.), Berlin, New York:Springer-Verlag, ISBN978-3-540-56963-3,MR0214602( 1st ed 1965) [[Mathematical Reviews|MR]]&nbsp;[http://www.ams.org/mathscinet-getitem?mr=0719371 0719371] (2nd ed) [[Mathematical Reviews|MR]]&nbsp;[http://www.ams.org/mathscinet-getitem?mr=1304906 1304906](3rd ed.)

• E.B. Vinberg, V.L. Popov, Invariant theory, in Algebraic geometry. IV. Encyclopaedia of Mathematical Sci-ences, 55 (translated from 1989 Russian edition) Springer-Verlag, Berlin, 1994. vi+284 pp. ISBN 3-540-54682-0

Chapter 12

Glossary of invariant theory

This page is a glossary of terms in invariant theory. For descriptions of particular invariant rings, see invariants ofa binary form, symmetric polynomials. For geometric terms used in invariant theory see the glossary of classicalalgebraic geometry. Definitions of many terms used in invariant theory can be found in (Sylvester 1853), (Cayley1860), (Burnside & Panton 1881), (Salmon 1885), (Elliot 1895), (Grace & Young 1903), (Glenn 1915), (Dolgachev2012), and the index to the fourth volume of Sylvester’s collected works includes many of the terms invented by him.Contents :

• Conventions

• !$@

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

28

12.1. CONVENTIONS 29

• T

• U

• V

• W

• XYZ

•

•

• See also

• References

12.1 Conventions-an Nouns ending in -an are often invariants named after people, as in Cayleyan, Hessian, Jacobian, Steinerian.

-ant Nouns ending in -ant are often invariants, as in determinant, covariant, and so on.

-ary Adjectives ending in -ary often refer to the number of variables of a form, as in unary, binary, ternary, quater-nary, quinary, senary, septenary, octonary, nonary, denary.

-ic Adjectives or nouns ending in -ic often refer to the degree of a form, as in linear or monic, quadric or quadratic,cubic, quartic or biquadratic, quintic, sextic, septic or septimic, octic or octavic, nonic, decic or decimic,undecic or undecimic, duodecic or duodecimic, and so on.

12.2 !$@(a0, a1, ..., an)(x,y)n Short for the form (n

0)a0xn + (n1)a1xn–1y+ ... + (nn)anyn. When the first ) has a circumflex or arrow on top of it, this means that the binomial coefficients areomitted. The parentheses are sometimes overlapped: (a0, . . . , an)(x, y)n

[] See Sylvester (1853, Glossary p. 543–548)

(αβγ...) The determinant of the matrix with entries αi, βi, γi,... For example, (αβ) means α1β2 – α2β1.

12.3 Aabsolute 1. The absolute invariant is essentially the j-invariant of an elliptic curve.

2. An absolute invariant is something fixed by a group action, in other words a (relative) invariant (something thattransforms according to a character) where the character is trivial.

allotrious See Sylvester (1853, Glossary p. 543–548), Archaic.

alternant 1. An archaic term for the commutator AB–BA of two operators A and B. (Elliott 1895, p.144)

2. An alternant matrix is a matrix such that the entries of each column are given by some fixed function of a variable.

annihilator An annihilator is a differential operator representing an element of a Lie algebra, so that invariants of agroup are killed by the annihilators. (Elliott 1895, p.108)

anti-invariant A relative invariant transforming according to a character of order 2 of a group such as the symmetricgroup.

30 CHAPTER 12. GLOSSARY OF INVARIANT THEORY

anti-seminvariant (Elliott 1895, p.126)

apocopated See Sylvester (1853, Glossary p. 543–548). Archaic.

Arf invariant Main article: Arf invariantAn invariant of quadratic forms over a field of order 2.

Aronhold invariant One of the two generators of degrees 4 and 6 of the ring of invariants of ternary cubic forms.(Dolgachev 2012, 3.1.1)

asyzygetic Linearly independent.

12.4 B

Bezoutiant Main article: BezoutiantA symmetric square matrix associated to two binary forms.

Bezoutic See Sylvester (1853, Glossary p. 543–548). Archaic.

Bezoutiod See Sylvester (1853, Glossary p. 543–548). Archaic.

bidegree An ordered pair of integers, giving the degrees of a form relative to two sets of variables.

biform A polynomial homogeneous in each of two sets of variables. In other words an element of SmV×SnW, usuallyconsidered as a representation of GLV×GLW.

binary Depending on 2 variables. Same as bivariate.

biquadratic Same as quartic, meaning degree 4.

biternary A biternary form is one in 6 variables, 3 transforming according to the fundamental representation of SL3

and 3 transforming according to its dual.

bivariate Depending on 2 variables. Same as binary.

Boolian invariant An invariant for the orthogonal group. (Elliott 1895, p.344)

bordered Hessian An alternative name for the reciprocant

bracket An invariant given by either the pairing of a vector and a vector in the dual space, or the determinant of amatrix form by n vectors of an n-dimensional space (in other words their exterior product in the top exteriorpower).

Brioschi covariant This is a degree 12 order 9 covariant of ternary cubic forms, introduced by Brioschi (1863).(Dolgachev 2012, 3.4.3)

12.5 C

canonical form A particularly simple representation of a form, such as a sum of powers of linear forms, or withmany zero coefficients. For example, the canonical form of a binary form of degree 2m+1 is a sum of m+1powers of linear forms.

canonisant

canonizant Main article: canonizantA covariant of a form, given by the catalecticant of the penultimate emanant. It is related to the canonical formof a form. For example, the canonizant of a binary form of degree 2n–1 has degree n and order n. (Elliott1895, p.21)

catalecticant Main article: catalecticantAn invariant vanishing on forms that are the sum of an unusually small number of powers of linear forms.

12.5. C 31

Cayley Ω process Main article: Cayley’s Ω processA certain differential operator used for constructing invariants.

Cayleyan Main article: CayleyanA contravariant.

characteristic See Sylvester (1853, Glossary p. 543–548)

class The class of a contravariant or concomitant is its degree in the covariant variables. See also degree and order.

Clebsch invariant (Dolgachev 2012, p.283)

co-Bezoutiant See Sylvester (1853, Glossary p. 543–548). Archaic.

cogredient Transforming according to the natural representation of a linear group. (Elliott 1895, p.55)

combinant A joint relative invariant of several forms of the same degree, that is unchanged if a multiple of one ofthe forms is added to another. Essentially a relative invariant of a product of two general linear groups. (Elliott1895, p.340) Sylvester (1853, Glossary p. 543–548) (Salmon 1885, p.161)

combinative Related to invariants of a product of groups. For example a combinative covariant is a covariant of aproduct of two groups.

commutant A generalization of the determinant to arrays of dimension greater than 2. (Cayley 1860)

complete A complete system of invariants is a set of generators for the ring of invariants.

concomitant A relative invariant of GL(V) acting on the polynomials over Sn(V)⊕V⊕V*.

conjunctive See Sylvester (1853, Glossary p. 543–548)

connex A form in two sets of variables, one set corresponding to a vector space and the other to its dual, or in otherwords an element of the symmetric algebra of V⊕V* for a vector space V. Introduced by Clebsch.

continuant Main article: Continuant (mathematics)A determinant of a tridiagonal matrix.(Salmon 1885, p.18)

contragredient Transforming according to the dual of the natural representation of a linear group. (Elliott 1895,p.74)

contravariant A relative invariant of GL(V) acting on the polynomials over Sn(V)⊕V.

convolution A method of constructing invariants from two other invariants. (Glenn 1915, p.87)

covariancy (Elliott 1895, p.83)

covariant 1. (Noun) A relative invariant of GL(V) acting on the polynomials over Sn(V)⊕V*.

2. (Adjective) Invariant under the action of a group, especially for functions between two spaces acted on by thegroup.

cross ratio The cross ratio is an invariant of 4 points of a projective line.

cubic (Adjective) Degree 3

(Noun) A form of degree 3

cubicovariant A covariant of degree 3, in particular an order 3 degree 3 covariant of a binary cubic given by theJacobian of the cubic and its Hessian.. (Elliott 1895, p.50)

cubinvariant An invariant of degree 3.

cubo- Used to form compound adjectives such as cubo-linear, cubo-quadric, and so on, indicating the bidegree ofsomething. For example, cubo-linear means having degree 3 in the first of two sets of variables and degree 1in the second.

cumulant The numerator or denominator of a continued fraction, often expressed as a determinant. Sylvester (1853,Glossary p. 543–548).

32 CHAPTER 12. GLOSSARY OF INVARIANT THEORY

12.6 Ddecic

decimic (Adjective) Degree 10

(Noun) A form of degree 10

degree 1. The degree of a form is the total power of the variables in it.

2. The degree of an invariant or covariant or contravariant means its degree in terms of the coefficients of the form.The degree of a form considered as a form is usually not its degree when considered as a covariant.

3. Some authors exchange the meanings of “degree” and “order” of a covariant or concomitant.

denary Depending on 10 variables

determinant The determinant is a joint invariant of n vectors of an n-dimensional space.

dialytic Sylvester’s dialytic method is a method for calculating resultants, essentially by expressing them as the de-terminant of a Sylvester matrix. See Sylvester (1853, Glossary p. 543–548). Archaic.

differentiant Another name for an invariant of a binary form. Archaic.

discriminant Main article: discriminantThe discriminant of a form in n variables is the multivariate resultant of the n differentials with respect to eachof the variables. For binary forms the discriminant vanishes if the form has multiple roots and is essentiallythe same as the discriminant of a polynomial of 1 variable. The discriminant of a form vanishes when thecorresponding hypersurface has singularities (as a scheme).

disjunctive See Sylvester (1853, Glossary p. 543–548)

divariant An alternative name for a concomitant suggested by Salmon (1885, p.121)

duodecic

duodecimic (Adjective) Degree 12

(Noun) A form of degree 12

12.7 Eeffective See Sylvester (1853, Glossary p. 543–548)

effluent See Sylvester (1853, Glossary p. 543–548). Archaic.

eliminant De Morgan’s name for the (multivariate) resultant, an invariant of n forms in n variables that vanishes ifthey have a common nonzero solution. (Elliott 1895, p.16)

emanant The rth emanant of a binary form in variables xi is a covariant given by the action of the rth power of thedifferential operator Σyi∂/∂xi. This is essentially the same as polarization. (Elliott 1895, p.56) Sylvester (1853,Glossary p. 543–548)

endoscopic See Sylvester (1853, Glossary p. 543–548). Archaic.

equianharmonic contravariant A weight 4 contravariant of binary quartics (Dolgachev 2012, 6.4)

evectant Main article: evectantA contravariant given by the action of an evector.

evector Main article: evectantA differential operator constructed from a binary form.

excess The excess of a polynomial in the coefficients a0,...ap of a form of degree p is ip–2w, where p is the degreeof the polynomial and w is its weight. (Elliott 1895, p.141)

12.8. F 33

exoscopic See Sylvester (1853, Glossary p. 543–548). Archaic.

extensor An element of the kth exterior power of a vector space that can be written as the exterior product of kvectors.

extent The extent of a polynomial in a0, a1,... is the largest value of p such that the polynomial involves ap. (Elliott1895, p.138)

12.8 F

facient One of the variables of a form (Cayley 1860)

facultative A facultative point is one where a given function is positive. (Salmon 1885, p.243)

form A homogeneous polynomial in several variables, also called a quantic.

functional determinant An archaic name for Jacobians

fundamental 1. The first fundamental theorem describes generators (called brackets) for the ring of invariantpolynomials on a sum of copies of a vector space V and its dual (for the special linear group of V). The secondfundamental theorem describes the syzygies between the generators.

2. For fundamental scale see Sylvester (1853, Glossary p. 543–548). Archaic.

3. A fundamental invariant is an element of a set of generators for a ring of invariants.

4. A fundamental system is a set of generators (for a ring of invariants, covariants, and so on).

12.9 G

Gordan Named for Paul Gordan.

1. Gordan’s theorem states that the ring of invariants of a binary form (or several binary forms) is finitely generated.

grade The highest power of a bracket factor in the symbolic expression for an invariant. (Glenn 1915, 4.8)

gradient A homogeneous polynomial in a0, ..., ap all of whose terms have the same weight, where an has weight n.(Elliott 1895, p.138) Archaic.

Gröbner basis Main article: Gröbner basisA basis for an ideal of a ring of polynomials chosen according to some rule to make computations easier.

ground form An element of a minimal set of homogeneous generators for the invariants of a form. Archaic.

12.10 H

hectic A joke term for a form of degree 100.

harmonic contravariant A weight 6 contravariant of binary quartics (Dolgachev 2012, 6.4)

harmonizant A bilinear invariant of two forms whose vanishing means they are polar. (Dolgachev 2012, p.75)

Hermite Named after Charles Hermite

1. The Hermite contravariant is a degree 12 class 9 contravariant of ternary cubics. (Dolgachev 2012, 3.4.3)

2. Hermite’s law of reciprocity states that the degree m covariants of a binary form of degree n correspond to thedegree n covariants of a binary form of degree m.

3. The Hermite invariant is the degree 18 skew invariant of a binary quintic.

34 CHAPTER 12. GLOSSARY OF INVARIANT THEORY

Hessian Main article: HessianA covariant of a form u, given by the determinant of the matrix with entries ∂2u/∂xi∂xj.

Hilbert Named after David Hilbert

A Hilbert series is a formal power series whose coefficients are dimensions of spaces of invariants of variousdegrees.

Hilbert’s theorem states that the ring of invariants of a finite-dimensional representation of a reductive group isfinitely generated.

homographic 1. A homographic transformation is a transformation taking x to (ax+b)/(cx+d).

2. A homographic relation between x and y is a relation of the form axy + bx + cy + d=0 .

hyperdeterminant Main article: hyperdeterminantAn invariant of a multidimensional array of coefficients, generalizing the determinant of a 2-dimensional array.

12.11 Iidentity covariant A form considered as a covariant of degree 1.

immanant Main article: immanant of a matrixA generalization of the determinant and permanent of a matrix

inertia The signature of a real quadratic form. See Sylvester (1853, Glossary p. 543–548)

integral rational function A polynomial.

intercalations See Sylvester (1853, Glossary p. 543–548). Archaic.

intermediate invariant An invariant of two forms constructed from two invariants of each of the forms. (Elliott1895, p.23)

intermutant A special form of permutant. (Cayley 1860)

invariant 1. (Adjective) Fixed by the action of a group

2. (Noun) An absolute invariant, meaning something fixed by a group action.

3. (Noun) A relative invariant, meaning something transforming according to a character of a group. In classicalinvariant theory it often refers to relatively invariant polynomials in the coefficients of a quantic, considered asa representation of a general linear group.

involutant See Sylvester’s collected papers, volume IV, page 135

irreducible Not expressible as a polynomial in things of smaller degree.

isobaric All terms having the same weight. (Elliott 1895, p.32)

12.12 JJacobian Main article: Jacobian matrix

A covariant of n forms fi in n variables xj, given by the determinant of the matrix with entries ∂fi/∂xj.

joint invariant A relative invariant for polynomials over reducible representation of a group, in particular a relativeinvariant for a several binary forms.

12.13 Kkenotheme Sylvester (1853, Glossary p. 543–548) defines this as “A finite system of discrete points defined by one

or more homogeneous equations in number one less than the number of variables contained therein.” This maymean an intersection of n hypersurfaces in n-dimensional projective space. Archaic.

12.14. L 35

12.14 Llinear Degree 1

lineo- Used to form compound adjectives such as lineo-linear, lineo-quadric, and so on, indicating the bidegree ofsomething. For example, lineo-linear means having degree 1 in each of two sets of variables. In particular thelineo-linear invariant of two binary forms has degree 1 in the coefficients of each form. (Elliott 1895, p.54)

Lüroth invariant A degree 54 invariant vanishing on Lüroth quartics (nonsingular quartic plane curves containingthe 10 vertices of a complete pentalateral). (Dolgachev 2012, p.295)

12.15 Mmeicatalecticizant Sylvester’s original term for what he later renamed the catalecticant. Archaic.

mixed concomitant A concomitant that involves both covariant and contravariant variables, in other words one thatis not a covariant or contravariant. (Elliott 1895, p.77)

modular Main article: Modular invariant of a groupDefined over a finite field.

modulus An alternative name for the determinant of a linear transformation. (Elliott 1895, p.3)

monic 1. Adjective. Having leading coefficient 1.

2. Adjective. Having degree 1.

3. Noun. A form of degree 1.

monotheme See Sylvester (1853, Glossary p. 543–548). Archaic.

12.16 Nnonary Depending on 9 variables

nonic (Adjective) Degree 9

(Noun) A form of degree 9

nullcone The cone of nullforms

nullform Main article: nullformA form on which all invariants with zero constant term vanish.

12.17 Ooctavic

octic (Adjective) Degree 8

(Noun) A form of degree 8

octonary Depending on 8 variables

Omega process Main article: Cayley’s Ω process

order 1. The degree of a covariant or concomitant in the variables of a form.

2. Some authors interchange the meaning of “degree” and “order” of a covaraint.

36 CHAPTER 12. GLOSSARY OF INVARIANT THEORY

3. See Sylvester (1853, Glossary p. 543–548)

ordinary An ordinary invariant means a relative invariant, in other words something transforming according to acharacter of a group, as opposed to an absolute invariant.

osculant Main article: osculantAn invariant of several forms of the same degree generalizing the tact-invariant of two forms, equal to thediscriminant if the number of forms is 1, and to the multivariate resultant if the number of forms is the numberof variables. Salmon (1885, p.171)

12.18 Ppartial transvectant Main article: transvectant

partition Main article: partition (number theory)An expression of a number as a sum of positive integers.(Elliott 1895, p.119)

peninvariant Same as seminvariant. (Cayley 1860)

permanent Main article: permanentA variation of the determinant of a matrix

permutant (Cayley 1860)

perpetuant Main article: perpetuantRoughly an irreducible covariant of a form of infinite order.

persymmetrical A persymmetrical matrix is a Hankel matrix. See Sylvester (1853, Glossary p. 543–548). Archaic.

Pfaffian Main article: PfaffianA square root of the determinant of a skew-symmetric matrix.

pippian An old name for the Cayleyan.

plagiogonal Related to or fixed by the orthogonal group of some quadratic form. See Sylvester’s collected papers,volume I, page 357

plexus A set of generators of an ideal, especially if the number of generators needed is larger than the codimensionof the corresponding variety.

polarization A method of reducing the degree of something by introducing extra variables.

principiant A reciprocant that is invariant under homographic substitutions, up to a constant facts. See Sylvester’scollected papers, vol IV, page 382

projective invariant 1. An invariant of the projective general linear group.

2. An invariant of a central extension of a group.

protomorph A set of protomorphs is a set of seminvariants, such that any seminvariant is a polynomial in theprotomorphs and the inverse of the first protomorph. (Elliott 1895, p.206)

12.19 Qquadratic

quadric (Adjective) Degree 2

(Noun) A form of degree 2

quadricovariant A covariant of degree 2. (Salmon 1885, p.261)

12.20. R 37

quadrinvariant An invariant of degree 2. Sylvester (1853, Glossary p. 543–548).

quadro- Degree 2. Used to form compound adjectives such as quadro-linear, quadro-quadric, and so on, indicatingthe bidegree of something. For example, quadro-linear means having degree 2 in the first of two sets ofvariables and degree 1 in the other.

quantic An archaic name for a homogeneous polynomial in several variables, now usually called a form.

quartic (Adjective) Degree 4

(Noun) A form of degree 4

quarticovariant A covariant of degree 4.

quartinvariant An invariant of degree 4

quarto- Used to form compound adjectives such as quarto-linear, quarto-quadric, and so on, indicating the bidegreeof something. For example, quarto-linear means having degree 4 in the first of two sets of variables and degree1 in the other.

quaternary Depending on 4 variables

quinary Depending on 5 variables.

quintic (Adjective) Degree 5

(Noun) A form of degree 5

quintinvariant An invariant of degree 5.

quippian Main article: quippian

12.20 R

rational integral function A polynomial.

reciprocal The reciprocal of a matrix is the adjugate matrix.

reciprocant 1. A contravariant of a ternary form, giving the equation of a dual curve. (Elliott 1895, p.400)

reciprocity Exchanging the degree of a form with the degree of an invariant. For example, Hermite’s law of reci-procity states that the degree p invariants of a form of degree n correspond to the degree n invariants of a formof degree p. (Elliott 1895, p.137)

reducible Expressible as a polynomial in things of smaller degree.

relative invariant Something transforming according to a 1-dimensional character of a group, often a power of thedeterminant. Same as ordinary invariant.

resultant 1.Main article: resultantA joint invariant of two binary forms that vanishes when they have a common root. More generally a (multi-variate) resultant is a joint invariant of n forms in n variables that vanishes if they have a common nontrivialzero. Sometimes called an eliminant in older books.

2. An archaic term for the determinant

revenant Suggested by Sylvester (collected works vol 3, page 593) as an alternative name for a perpetuant.

Reynolds operator Main article: Reynolds operatorProjection onto the fixed vectors

rhizoristic See Sylvester (1853, Glossary p. 543–548). Archaic.

38 CHAPTER 12. GLOSSARY OF INVARIANT THEORY

12.21 S

Salmon invariant A degree 60 invariant vanishing on ternary quartics with an inflection bitangent. (Dolgachev2012, 6.4)

Scorza covariant A covariant of ternary quartics. (Dolgachev 2012, 6.3.4)

semicovariant An analogue of seminvariants for covariants. See (Burnside & Panton 1881, p.329)

semi-invariant

seminvariant 1. The leading term of a covariant, also called its source. (Grace & Young 1903, section 33)

2. An invariant of the group of upper triangular matrices.

senary Depending on 6 variables. (Rare)

septenary Depending on 7 variables

septic

septimic (Adjective) Degree 7

(Noun) A form of degree 7

sextic (Adjective) Degree 6

(Noun) A form of degree 6

sexticovariant A covariant of degree 6

sextinvariant An invariant of degree 6 (Salmon 1885, p.262)

signaletic See Sylvester (1853, Glossary p. 543–548). Archaic.

singular 1. See Sylvester (1853, Glossary p. 543–548)

skew A skew invariant is a relative invariant of a group G that changes sign under an element of order 2 in itsabelianization. In particular for the general linear group it changes sign under elements of determinant –1, andfor the symmetric group it changes sign under odd permutations. For binary forms skew invariants are theinvariants of odd weight. They do not exist for binary quadrics, cubics, or quartics, but do for binary quintics.(Elliott 1895, p.112)

source The source of a covariant is its leading term, when the covariant is considered as a form. Also called aseminvariant. (Elliott 1895, p.126)

Steinerian Main article: Steinerian

symbolic Main article: symbolic methodThe symbolicmethod is a way of representing invariants, that repeatedly uses the identification of the symmetricpower of a vector space with the symmetric elements of a tensor power.

syrrhizoristic Sylvester (1853, Glossary p. 543–548) defined this as “A syrrhizoristic series is a series of discon-nected functions which serve to determine the effective intercalations of the real roots of two functions lyingbetween any assigned limits.” Archaic. This term does not seem to have been used (or understood) by anyoneother than Sylvester.

syzygant (Elliott 1895, p.198)

syzygetic See Sylvester (1853, Glossary p. 543–548)

syzygy A linear or algebraic relation, especially one between generators of a ring or module.

12.22. T 39

12.22 Ttacinvariant

tact invariant An invariant of one or two ternary forms that vanishes if the corresponding curve touches itself, or ifthe two curves touch each other. It is generalized by the osculant.

tamisage Sylvester’s name for his method of guessing the degrees of a generating set of invariants or covariants byexamining the generating function.(Elliott 1895, p.175). Archaic.

tantipartite An archaic term for multilinear. (Cayley 1860)Tschirnhaus transformation Main article: Tschirnhaus transformation

ternary Depending on 3 variablesToeplitz invariant An invariant of nets of quadrics in 3-dimensional projective space that vanishes on nets with a

common polar pentahedron. (Dolgachev 2012, p.51)transfer A method of constructing contravariants of forms in n+1 variables from invariants of forms in n variables.

(Dolgachev 2012, 3.4.2)transvectant Main article: transvectant

An invariant formed from n invariants in n variables using Cayley’s omega process. (Elliott 1895, p.71)trinomial A polynomial with at most three non-zero coefficients.

12.23 Uueberschiebung Transvectant. (Elliott 1895, p.171)umbrae

umbral See Sylvester (1853, Glossary p. 543–548)unary Depending on 1 variable. Same as univariate.undecic

undecimic (Adjective) Degree 11(Noun) A form of degree 11unimodular Having determinant 1unitarian trick Finite-dimensional representations of a semisimple Lie group are equivalent to finite-dimensional

representations of a compact form, and are therefore completely reducible.univariate Depending on 1 variable. Same as unary.universal concomitant The pairing between a vector space and its dual, considered as a concomitant. (Elliott 1895,

p.77)

12.24 V

12.25 Wweight 1. The power of the determinant appearing in the formula for transformation of a relative invariant.2. A character of a torus3. See Sylvester (1853, Glossary p. 543–548)4. The weight of ai is i, and the weight of a product of monomials is the sum of their weights.

40 CHAPTER 12. GLOSSARY OF INVARIANT THEORY

12.26 XYZzeta

ζ A product of squared differences. See Sylvester (1853, Glossary p. 543–548)

12.27 See also• Glossary of classical algebraic geometry

12.28 References• Burnside, William Snow; Panton, Arthur William (1881), The theory of equations: With an introduction to thetheory of binary algebraic forms, 2 volumes, Hodges, Figgis & co., MR 0115987

• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525 Reprinted as Dieudonné, JeanA.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

• Cayley, Arthur (1860), “Recent terminology in mathematics”, The English Cyclopaedia 5, Charles Knight,London, pp. 534–542, Reprinted in Cayley’s collected works, volume IV, pages 594–608

• Crilly, Tony (2006), Arthur Cayley. Mathematician laureate of the Victorian age, Johns Hopkins UniversityPress, ISBN 978-0-8018-8011-7, MR 2284396

• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series 296,Cambridge University Press, doi:10.1017/CBO9780511615436, ISBN 978-0-521-52548-0, MR 2004511

• Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press,ISBN 978-1-107-01765-8

• Elliott, Edwin Bailey (1895), An introduction to the algebra of quantics, Oxford, Clarendon Press, Reprintedby Chelsea Scientific Books 1964

• Glenn, Oliver E. (1915), A Treatise on the Theory of Invariants, Ginn and company, ISBN 978-1-4297-0030-6

• Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge: Cambridge University Press

• Hilbert, David (1890), “Ueber die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831

• Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen 42 (3):313, doi:10.1007/BF01444162

• Olver, Peter J. (1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 0-521-55821-2

• Salmon, George (1885) [1859], Lessons introductory to the modern higher algebra (4th ed.), Dublin, Hodges,Figgis, and Co., ISBN 978-0-8284-0150-0

• Sylvester, James Joseph (1853), “On a Theory of the Syzygetic Relations of Two Rational Integral Functions,Comprising an Application to the Theory of Sturm’s Functions, and That of the Greatest Algebraical CommonMeasure”, Philosophical Transactions of the Royal Society of London (The Royal Society) 143: 407–548,doi:10.1098/rstl.1853.0018, ISSN 0080-4614, JSTOR 108572

• Sylvester, James Joseph; Franklin, F. (1879), “Tables of the Generating Functions and Groundforms for theBinary Quantics of the First Ten Orders”, American Journal of Mathematics (The Johns Hopkins UniversityPress) 2 (3): 223–251, doi:10.2307/2369240, ISSN 0002-9327

• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255

12.29. EXTERNAL LINKS 41

12.29 External links• Brouwer, Andries E., Invariants of binary forms

Chapter 13

Gram’s theorem

In mathematics, Gram’s theorem states that an algebraic set in a finite-dimensional vector space invariant undersome linear group can be defined by absolute invariants. (Dieudonné & Carrell 1970, p. 31). It is named after J. P.Gram, who published it in 1874.

13.1 References• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525. Reprinted by Academic Press(1971), MR 0279102.

• Gram, J. P. (1874), “Sur quelques théorèmes fondamentaux de l'algèbre moderne”, Mathematische Annalen(in French) 7: 230–240, doi:10.1007/bf02104801.

42

Chapter 14

Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computationalcommutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over afield K[x1, ..,xn]. A Gröbner basis allows many important properties of the ideal and the associated algebraic varietyto be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation isone of the main practical tools for solving systems of polynomial equations and computing the images of algebraicvarieties under projections or rational maps.Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid’s algorithm forcomputing polynomial greatest common divisors, and Gaussian elimination for linear systems.[1]

Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger’s algorithm), byBruno Buchberger in his Ph.D. thesis. He named them after his advisor Wolfgang Gröbner. In 2007, Buchbergerreceived the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work.However, the Russian mathematician N. M. Gjunter had introduced a similar notion in 1913, published in variousRussian mathematical journals. These papers were largely ignored by the mathematical community until their redis-covery in 1987 by Bodo Renschuch et al.[2] An analogous concept for local rings was developed independently byHeisuke Hironaka in 1964, who named them standard bases.The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized toother structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras.

14.1 Background

14.1.1 Polynomial ring

Main article: Polynomial ring

Gröbner bases are primarily defined for ideals in a polynomial ring R = K[x1, . . . , xn] over a field K. Although thetheory works for any field, most Gröbner basis computations are done either when K is the field of rationals or theintegers modulo a prime number.A polynomial is a sum c1M1 + · · · + cmMm where the ci are nonzero elements of K and the Mi are monomialsor “power products” of the variables. This means that a monomial M is a productM = xa11 · · ·xann , where the aiare nonnegative integers. The vector A = [a1, . . . , an] is called the exponent vector of M. The notation is oftenabbreviated as xa11 · · ·xann = XA . Monomials are uniquely defined by their exponent vectors so computers canrepresent monomials efficiently as exponent vectors, and polynomials as lists of exponent vectors and their coefficients.If F = f1, . . . , fk is a finite set of polynomials in a polynomial ring R, the ideal generated by F is the set of linearcombinations of elements from F with coefficients in all of R:

43

44 CHAPTER 14. GRÖBNER BASIS

⟨f1, . . . , fk⟩ =

k∑i=1

gifi | g1, . . . , gk ∈ K[x1, . . . , xn]

.

14.1.2 Monomial ordering

Main article: Monomial order

All operations related to Gröbner bases require the choice of a total order on the monomials, with the followingproperties of compatibility with multiplication. For all monomials M, N, P,

1. M < N ⇐⇒MP < NP

2. M < MP .

A total order satisfying these condition is sometimes called an admissible ordering.These conditions imply Noetherianity, which means that every strictly decreasing sequence of monomials is finite.Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, threemonomial orderings are specially important for the applications:

• Lexicographical ordering, commonly called lex or plex (for pure lexical ordering).

• Total degree reverse lexicographical ordering, commonly called degrevlex.

• Elimination ordering, lexdeg.

Gröbner basis theory was initially introduced for the lexicographical ordering. It was soon realised that the Gröbnerbasis for degrevlex is almost always much easier to compute, and that it is almost always easier to compute a lex Gröb-ner basis by first computing the degrevlex basis and then using a “change of ordering algorithm”. When eliminationis needed, degrevlex is not convenient; both lex and lexdeg may be used but, again, many computations are relativelyeasy with lexdeg and almost impossible with lex.Once a monomial ordering is fixed, the terms of a polynomial (product of a monomial with its nonzero coefficient)are naturally ordered by decreasing monomials (for this order). This makes the representation of a polynomial as anordered list of pairs coefficient–exponent vector a canonical representation of the polynomials. The first (greatest)term of a polynomial p for this ordering and the corresponding monomial and coefficient are respectively called theleading term, leading monomial and leading coefficient and denoted, in this article, lt(p), lm(p) and lc(p).

14.1.3 Reduction

The concept of reduction, also called multivariate division or normal form computation, is central to Gröbnerbasis theory. It is a multivariate generalization of the Euclidean division of univariate polynomials.In this section we suppose a fixed monomial ordering, which will not be defined explicitly.Given two polynomials f and g, one says that f is reducible by g if some monomial m in f is a multiple of the leadingmonomial lm(g) of g. If m happens to be the leading monomial of f then one says that f is lead-reducible by g. If cis the coefficient of m in f and m = q lm(g), the one-step reduction of f by g is the operation that associates to f thepolynomial

red1(f, g) = f − c

lc(g) q g.

The main properties of this operation are that the resulting polynomial does not contain the monomial m and that themonomials greater than m (for the monomial ordering) remain unchanged. This operation is not, in general, uniquelydefined; if several monomials in f are multiples of lm(g) one may choose arbitrarily the one that is reduced. In

14.2. FORMAL DEFINITION 45

practice, it is better to choose the greatest one for the monomial ordering, because otherwise subsequent reductionscould reintroduce the monomial that has just been removed.Given a finite set G of polynomials, one says that f is reducible or lead-reducible by G if it is reducible or lead-reducible, respectively, by an element ofG. If it is the case, then one defines red1(f,G) = red1(f, g) . The (complete)reduction of f by G consists in applying iteratively this operator red1 until getting a polynomial red(f,G) , whichis irreducible by G. It is called a normal form of f by G. In general this form is not uniquely defined (this is not acanonical form); this non-uniqueness is the starting point of Gröbner basis theory.For Gröbner basis computations, except at the end, it is not necessary to do a complete reduction: a lead-reduction issufficient, which saves a large amount of computation.The definition of the reduction shows immediately that, if h is a normal form of f by G, then we have

f = h+∑g∈G

qg g,

where the qg are polynomials. In the case of univariate polynomials, if G is reduced to a single element g, then h isthe remainder of the Euclidean division of f by g, qg is the quotient and the division algorithm is exactly the processof lead-reduction. For this reason, some authors use the term multivariate division instead of reduction.

14.2 Formal definition

A Gröbner basis G of an ideal I in a polynomial ring R over a field is characterized by any one of the followingproperties, stated relative to some monomial order:

• the ideal given by the leading terms of polynomials in I is itself generated by the leading terms of the basis G;

• the leading term of any polynomial in I is divisible by the leading term of some polynomial in the basis G;

• multivariate division of any polynomial in the polynomial ring R by G gives a unique remainder;

• multivariate division of any polynomial in the ideal I by G gives remainder 0.

All these properties are equivalent; different authors use different definitions depending on the topic they choose.The last two properties allow calculations in the factor ring R/I with the same facility as modular arithmetic. It is asignificant fact of commutative algebra that Gröbner bases always exist, and can be effectively obtained for any idealstarting with a generating subset.Multivariate division requires a monomial ordering, the basis depends on the monomial ordering chosen, and dif-ferent orderings can give rise to radically different Gröbner bases. Two of the most commonly used orderings arelexicographic ordering, and degree reverse lexicographic order (also called graded reverse lexicographic order or sim-ply total degree order). Lexicographic order eliminates variables, however the resulting Gröbner bases are often verylarge and expensive to compute. Degree reverse lexicographic order typically provides for the fastest Gröbner basiscomputations. In this order monomials are compared first by total degree, with ties broken by taking the smallestmonomial with respect to lexicographic ordering with the variables reversed.In most cases (polynomials in finitely many variables with complex coefficients or, more generally, coefficients overany field, for example), Gröbner bases exist for any monomial ordering. Buchberger’s algorithm is the oldest andmost well-known method for computing them. Other methods are the Faugère’s F4 and F5 algorithms, based on thesamemathematics as the Buchberger algorithm, and involutive approaches, based on ideas from differential algebra.[3]There are also three algorithms for converting a Gröbner basis with respect to one monomial order to a Gröbner basiswith respect to a different monomial order: the FGLM algorithm, the Hilbert Driven Algorithm and the Gröbner walkalgorithm. These algorithms are often employed to compute (difficult) lexicographic Gröbner bases from (easier) totaldegree Gröbner bases.A Gröbner basis is termed reduced if the leading coefficient of each element of the basis is 1 and no monomial inany element of the basis is in the ideal generated by the leading terms of the other elements of the basis. In the worstcase, computation of a Gröbner basis may require time that is exponential or even doubly exponential in the numberof solutions of the polynomial system (for degree reverse lexicographic order and lexicographic order, respectively).

46 CHAPTER 14. GRÖBNER BASIS

Despite these complexity bounds, both standard and reduced Gröbner bases are often computable in practice, andmost computer algebra systems contain routines to do so.The concept and algorithms of Gröbner bases have been generalized to submodules of free modules over a polynomialring. In fact, if L is a free module over a ring R, then one may consider R⊕L as a ring by defining the product of twoelements of L to be 0. This ring may be identified withR[e1, . . . , el]/ ⟨eiej |1 ≤ i ≤ j ≤ l⟩ , where e1, . . . , el is abasis of L. This allows to identify a submodule of L generated by g1, . . . , gk with the ideal ofR[e1, . . . , el] generatedby g1, . . . , gk and the products eiej , 1 ≤ i ≤ j ≤ l . If R is a polynomial ring, this reduces the theory and thealgorithms of Gröbner bases of modules to the theory and the algorithms of Gröbner bases of ideals.The concept and algorithms of Gröbner bases have also been generalized to ideals over various rings, commutativeor not, like polynomial rings over a principal ideal ring or Weyl algebras.

14.3 Example and counterexample

Y

x^2 - y = 0

x^3 - x = 0

X(0,0)

(1,1)(-1,1)

The zeroes of f(x,y) form the red parabola; the zeroes of g(x,y) form the three blue vertical lines. Their intersection consists of threepoints.

LetR =Q[x,y] be the ring of bivariate polynomials with rational coefficients and consider the ideal I = <f,g> generatedby the polynomialsf(x,y) = x2 - yg(x,y) = x3 - xTwo other elements of I are the polynomialsh(x,y) = -(x2 + y - 1)f(x,y) + x.g(x,y) = y2 - yk(x,y) = -x.f(x,y) + g(x,y) = xy - xUnder lexicographic ordering with x > y we havelt(f) = x2

lt(g) = x3

lt(h) = y2

14.4. PROPERTIES AND APPLICATIONS OF GRÖBNER BASES 47

The ideal generated by lt(f),lt(g) only contains polynomials that are divisible by x2 which excludes lt(h) = y2; itfollows that f, g is not a Gröbner basis for I.On the other hand we can show that f, k, h is indeed a Gröbner basis for I.First note that f and g, and therefore also h, k and all the other polynomials in the ideal I have the following threezeroes in the (x,y) plane in common, as indicated in the figure: (1,1),(−1,1),(0,0). Those three points are notcollinear, so I does not contain any polynomial of the first degree. Neither can I contain any polynomials of thespecial formm(x,y) = cx + p(y)with c a nonzero rational number and p a polynomial in the variable y only; the reason being that such an m can neverhave two distinct zeroes with the same value for y (in this case, the points (1,1) and (−1,1)).From the above it follows that I, apart from the zero polynomial, only contains polynomials whose leading term hasdegree greater than or equal to 2; therefore their leading terms are divisible by at least one of the three monomials x2,xy, y2 = lt(f),lt(k),lt(h). This means that f, k, h is a Gröbner basis for I with respect to lexicographic orderingwith x > y.

14.4 Properties and applications of Gröbner bases

Unless explicitly stated, all the results that follow[4] are true for any monomial ordering (see that article for thedefinitions of the different orders that are mentioned below).It is a common misconception to think that the lexicographical order is needed for some of these results. On thecontrary, the lexicographical order is, almost always, the most difficult to compute, and using it makes unpracticalmany computations that are relatively easy with graded reverse lexicographic order (grevlex), or, when elimination isneeded, the elimination order (lexdeg) which restricts to grevlex on each block of variables.

14.4.1 Equality of ideals

Reduced Gröbner bases are unique for any given ideal and any monomial ordering. Thus two ideals are equal ifand only if they have the same (reduced) Gröbner basis (usually a Gröbner basis software always produces reducedGröbner bases).

14.4.2 Membership and inclusion of ideals

The reduction of a polynomial f by the Gröbner basis G of an ideal I yields 0 if and only if f is in I. This allows totest the membership of an element in an ideal. Another method consists in verifying that the Gröbner basis of G∪fis equal to G.To test if the ideal I generated by f1, ...,fk is contained in the ideal J, it suffices to test that every fi is in J. One mayalso test the equality of the reduced Gröbner bases of J and J∪f1, ...,fk.

14.4.3 Solutions of a system of algebraic equations

Main article: System of polynomial equations

Any set of polynomials may be viewed as a system of polynomial equations by equating the polynomials to zero. Theset of the solutions of such a system depends only on the generated ideal, and, therefore does not change when thegiven generating set is replaced by the Gröbner basis, for any ordering, of the generated ideal. Such a solution, withcoordinates in an algebraically closed field containing the coefficients of the polynomials, is called a zero of the ideal.In the usual case of rational coefficients, this algebraically closed field is chosen as the complex field.An ideal does not have any zero (the system of equations is inconsistent) if and only if 1 belongs to the ideal (this isHilbert’s Nullstellensatz), or, equivalently, if its Gröbner basis (for any monomial ordering) contains 1, or, also, if thecorresponding reduced Gröbner basis is [1].

48 CHAPTER 14. GRÖBNER BASIS

Given the Gröbner basis G of an ideal I, it has only a finite number of zeros, if and only if, for each variable x,G contains a polynomial with a leading monomial that is a power of x (without any other variable appearing in theleading term). If it is the case the number of zeros, counted with multiplicity, is equal to the number of monomialsthat are not multiple of any leading monomial of G. This number is called the degree of the ideal.When the number of zeros is finite, the Gröbner basis for a lexicographical monomial ordering provides, theoreticallya solution: the first coordinates of a solution is a root of the greatest common divisor of polynomials of the basis thatdepends only of the first variable. After substituting this root in the basis, the second coordinates of this solution is aroot of the greatest common divisor of the resulting polynomials that depends only on this second variable, and so on.This solving process is only theoretical, because it implies GCD computation and root-finding of polynomials withapproximate coefficients, which are not practicable because of numeric instability. Therefore, other methods havebeen developed to solve polynomial systems through Gröbner bases (see System of polynomial equations for moredetails).

14.4.4 Dimension, degree and Hilbert series

The dimension of an ideal I in a polynomial ring R is the Krull dimension of the ring R/I and is equal to the dimensionof the algebraic set of the zeros of I. It is also equal to number of hyperplanes in general position which are neededto have an intersection with the algebraic set, which is a finite number of points. The degree of the ideal and of itsassociated algebraic set is the number of points of this finite intersection, counted with multiplicity. In particular, thedegree of an hypersurface is equal to the degree of its definition polynomial.Both degree and dimension depends only on the set of the leading monomials of the Gröbner basis of the ideal forany monomial ordering.The dimension is the maximal size of a subset S of the variables such that there is no leading monomial dependingonly on the variables in S. Thus, if the ideal has dimension 0, then for each variable x there is a leading monomial inthe Gröbner basis that is a power of x.Both dimension and degree may be deduced from the Hilbert series of the ideal, which is the series

∑∞i=0 dit

i , wheredi is the number of monomials of degree i that are not multiple of any leading monomial in the Gröbner basis. TheHilbert series may be summed into a rational fraction

∞∑i=0

diti =

P (t)

(1− t)d,

where d is the dimension of the ideal and P (t) is a polynomial such that P (1) is the degree of the ideal.Although the dimension and the degree do not depend on the choice of the monomial ordering, the Hilbert series andthe polynomial P (t) change when one changes of monomial ordering.Most computer algebra systems that provide functions to compute Gröbner bases provide also functions for computingthe Hilbert series, and thus also the dimension and the degree.

14.4.5 Elimination

The computation of Gröbner bases for an elimination monomial ordering allows computational elimination theory.This is based on the following theorem.Let us consider a polynomial ring K[x1, . . . , xn, y1, . . . , ym] = K[X,Y ], in which the variables are split into twosubsets X and Y. Let us also choose an elimination monomial ordering “eliminating” X, that is a monomial orderingfor which two monomials are compared by comparing first the X-parts, and, in case of equality only, considering theY-parts. This implies that a monomial containing an X-variable is greater than every monomial independent of X. IfG is a Gröbner basis of an ideal I for this monomial ordering, then G ∩K[Y ] is a Gröbner basis of I ∩K[Y ] (thisideal is often called the elimination ideal). Moreover, a polynomial belongs to G ∩ K[Y ] if and only if its leadingterm belongs toG∩K[Y ] (this makes very easy the computation ofG∩K[Y ] , as only the leading monomials needto be checked).This elimination property has many applications, some of them are reported in the next sections.

14.4. PROPERTIES AND APPLICATIONS OF GRÖBNER BASES 49

Another application, in algebraic geometry, is that elimination realizes the geometric operation of projection of anaffine algebraic set into a subspace of the ambient space: with above notation, the (Zariski closure of) the projectionof the algebraic set defined by the ideal I into the Y-subspace is defined by the ideal I ∩K[Y ].

The lexicographical ordering such thatx1 > · · · > xn is an elimination ordering for every partition x1, . . . , xk, xk+1, . . . , xn.Thus a Gröbner basis for this ordering carries much more information than usually necessary. This may explain whyGröbner bases for the lexicographical ordering are usually the most difficult to compute.

14.4.6 Intersecting ideals

If I and J are two ideals generated respectively by f1, ..., fm and g1, ..., gk, then a single Gröbner basis computa-tion produces a Gröbner basis of their intersection I ∩ J. For this, one introduces a new indeterminate t, and one usesan elimination ordering such that the first block contains only t and the other block contains all the other variables(this means that a monomial containing t is greater than every monomial that do not contain t. With this monomialordering, a Gröbner basis of I ∩ J consists in the polynomials that do not contain t, in the Gröbner basis of the ideal

K = ⟨t2, tf1, . . . , tfm, (1− t)g1, . . . , (1− t)gk⟩.

In other words, I ∩ J is obtained by eliminating t in K. This may be proven by remarking that the ideal K consists inthe polynomials (a − b)t + b such that a ∈ I and b ∈ J . Such a polynomial is independent of t if and only a=b,which means that b ∈ I ∩ J.

14.4.7 Implicitization of a rational curve

A rational curve is an algebraic curve that has a parametric equation of the form

x1 =f1(t)

g1(t)

...

xn =fn(t)

gn(t),

where fi(t) and gi(t) are univariate polynomials for 1 ≤ i ≤ n. One may (and will) suppose that fi(t) and gi(t) arecoprime (they have no non-constant common factors).Implicitization consists in computing the implicit equations of such a curve. In case of n = 2, that is for plane curves,this may be computed with the resultant. The implicit equation is the following resultant:

Rest(g1x1 − f1, g2x2 − f2).

Elimination with Gröbner bases allow to implicitize for any value of n, simply by eliminating t in the ideal ⟨g1x1 −f1, . . . , gnx2 − fn⟩. If n = 2, the result is the same as with the resultant, if the map t 7→ (x1, x2) is injective foralmost every t. In the other case, the resultant is a power of the result of the elimination.

14.4.8 Saturation

When modeling a problem by polynomial equations, it is highly frequent that some quantities are supposed to be nonzero, because, if they are zero, the problem becomes very different. For example, when dealing with triangles, manyproperties become false if the triangle is degenerated, that is if the length of one side is equal to the sum of the lengthsof the other sides. In such situations, there is no hope to deduce relevant information from the polynomial system ifthe degenerate solutions are not dropped out. More precisely, the system of equations defines an algebraic set whichmay have several irreducible components, and one has to remove the components on which the degeneracy conditionsare everywhere zero.This is done by saturating the equations by the degeneracy conditions, which may be done by using the eliminationproperty of Gröbner bases.

50 CHAPTER 14. GRÖBNER BASIS

Definition of the saturation

The localization of a ring consists in adjoining to it the formal inverses of some elements. This section concernsonly the case of a single element, or equivalently a finite number of elements (adjoining the inverses of severalelements is equivalent to adjoin the inverse of their product. The localization of a ring R by an element f is the ringRf = R[t]/(1 − ft), where t is a new indeterminate representing the inverse of f. The localization of an ideal Iof R is the ideal RfI of Rf .When R is a polynomial ring, computing in Rf is not efficient, because of the need tomanage the denominators. Therefore, the operation of localization is usually replaced by the operation of saturation.The saturation with respect to f of an ideal I in R is the inverse image of RfI under the canonical map from R toRf . It is the ideal I : f∞ = g ∈ R|(∃k ∈ N)fkg ∈ I consisting in all elements of R whose products by somepower of f belongs to I.If J is the ideal generated by I and 1-ft in R[t], then I : f∞ = J ∩ R. It follows that, if R is a polynomial ring,a Gröbner basis computation eliminating t allows to compute a Gröbner basis of the saturation of an ideal by apolynomial.The important property of the saturation, which ensures that it removes from the algebraic set defined by the ideal Ithe irreducible components on which the polynomial f is zero is the following: The primary decomposition of I : f∞

consists in the components of the primary decomposition of I that do not contain any power of f.

Computation of the saturation

A Gröbner basis of the saturation by f of a polynomial ideal generated by a finite set of polynomials F, may beobtained by eliminating t in F ∪ 1− tf, that is by keeping the polynomials independent of t in the Gröbner basisof F ∪ 1− tf for an elimination ordering eliminating t.Instead of using F, one may also start from a Gröbner basis of F. It depends on the problems, which method is mostefficient. However, if the saturation does not remove any component, that is if the ideal is equal to its saturatedideal, computing first the Gröbner basis of F is usually faster. On the other hand if the saturation removes somecomponents, the direct computation may be dramatically faster.If one want to saturate with respect to several polynomials f1, . . . , fk or with respect to a single polynomial whichis a product f = f1 . . . fk, there are three ways to proceed, that give the same result but may have very differentcomputation times (it depends on the problem which is the most efficient).

• Saturating by f = f1 . . . fk in a single Gröbner basis computation.

• Saturating by f1, then saturating the result by f2, and so on.

• Adding to F or to its Gröbner basis the polynomials 1− t1f1, . . . , 1− tkfk, and eliminating the ti in a singleGröbner basis computation.

14.4.9 Effective Nullstellensatz

Hilbert’s Nullstellensatz has two versions. The first one asserts that a set of polynomials has an empty set of commonzeros in an algebraic closure of the field of the coefficients if and only if 1 belongs to the generated ideal. This iseasily tested with a Gröbner basis computation, because 1 belongs to an ideal if and only if 1 belongs to the Gröbnerbasis of the ideal, for any monomial ordering.The second version asserts that the set of common zeros (in an algebraic closure of the field of the coefficients) of anideal is contained in the hypersurface of the zeros of a polynomial f, if and only if a power of f belongs to the ideal.This may be tested by a saturating the ideal by f; in fact, a power of f belongs to the ideal if and only if the saturationby f provides a Gröbner basis containing 1.

14.4.10 Implicitization in higher dimension

By definition, an affine rational variety of dimension k may be described by parametric equations of the form

14.5. IMPLEMENTATIONS 51

x1 =p1p0

...

xn =pnp0,

where p0, . . . , pn are n+1 polynomials in the k variables (parameters of the parameterization) t1, . . . , tk. Thus theparameters t1, . . . , tk and the coordinates x1, . . . , xn of the points of the variety are zeros of the ideal

I = ⟨p0x1 − p1, . . . , p0xn − pn⟩ .

One could guess that it suffices to eliminate the parameters to obtain the implicit equations of the variety, as it hasbeen done in the case of curves. Unfortunately this is not always the case. If the pi have a common zero (sometimescalled base point), every irreducible component of the non-empty algebraic set defined by the pi is an irreduciblecomponent of the algebraic set defined by I. It follows that, in this case, the direct elimination of the ti provides anempty set of polynomials.Therefore, if k>1, two Gröbner basis computations are needed to implicitize:

1. Saturate I by p0 to get a Gröbner basis G

2. Eliminate the ti from G to get a Gröbner basis of the ideal (of the implicit equations) of the variety.

14.5 Implementations• CoCoA free computer algebra system for computing Gröbner bases.

• GAP free computer algebra system that can perform Gröbner bases calculations.

• FGb, Faugère’s own implementation of his F4 algorithm, available as a Maple library.[5] To the date, as of2014, it is, with Magma, the fastest implementation for rational coefficients and coefficients in a finite field ofprime order

• Macaulay 2 free software for doing polynomial computations, particularly Gröbner bases calculations.

• Magma has a very fast implementation of the Faugère’s F4 algorithm.[6]

• Maple has implementations of the Buchberger and Faugère F4 algorithms, as well as Gröbner trace

• Mathematica includes an implementation of the Buchberger algorithm, with performance-improving tech-niques such as the Gröbner walk, Gröbner trace, and an improvement for toric bases

• SINGULAR free software for computing Gröbner bases

• Sage provides a unified interface to several computer algebra systems (including SINGULAR and Macaulay),and includes a few Gröbner basis algorithms of its own.

• SymPy Python computer algebra system uses Gröbner bases to solve polynomial systems

14.6 See also• Buchberger’s algorithm

• Faugère’s F4 and F5 algorithms

• Graver basis

• Gröbner–Shirshov basis

• Regular chains, an alternative way to represent algebraic sets

52 CHAPTER 14. GRÖBNER BASIS

14.7 References[1] Lazard, D. (1983). “Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations”. Computer

Algebra. Lecture Notes in Computer Science 162. pp. 146–156. doi:10.1007/3-540-12868-9_99. ISBN 978-3-540-12868-7.

[2] Bodo Renschuch, Hartmut Roloff, Georgij G. Rasputin et. al. (2003). Contributions to Constructive Polynomial IdealTheory XXIII: Forgotten Works of Leningrad Mathematician N. M. Gjunter on Polynomial Ideal Theory. ACM SIGSAMBulletin, Vol 37, No 2, (2003): 35–48. English translation by Michael Abramson.

[3] Vladimir P. Gerdt, Yuri A. Blinkov (1998). Involutive Bases of Polynomial Ideals, Mathematics and Computers in Simu-lation, 45:519ff

[4] David Cox, John Little, and Donal O'Shea (1997). Ideals, Varieties, and Algorithms: An Introduction to ComputationalAlgebraic Geometry and Commutative Algebra. Springer. ISBN 0-387-94680-2.

[5] FGb home page

[6] Allan Steel’s Gröbner Basis Timings Page

14.8 Further reading

• William W. Adams, Philippe Loustaunau (1994). An Introduction to Gröbner Bases. American MathematicalSociety, Graduate Studies in Mathematics, Volume 3. ISBN 0-8218-3804-0

• Huishi Li (2011). Gröbner Bases in Ring Theory. World Scientific Publishing, ISBN 978-981-4365-13-0

• Thomas Becker, Volker Weispfenning (1998). Gröbner Bases. Springer Graduate Texts in Mathematics 141.ISBN 0-387-97971-9

• Bruno Buchberger (1965). An Algorithm for Finding the Basis Elements of the Residue Class Ring of a ZeroDimensional Polynomial Ideal. Ph.D. dissertation, University of Innsbruck. English translation by MichaelAbramson in Journal of Symbolic Computation 41 (2006): 471–511. [This is Buchberger’s thesis inventingGröbner bases.]

• Bruno Buchberger (1970). An Algorithmic Criterion for the Solvability of a System of Algebraic Equations.AequationesMathematicae 4 (1970): 374–383. English translation byMichael Abramson and Robert Lumbertin Gröbner Bases and Applications (B. Buchberger, F. Winkler, eds.). London Mathematical Society LectureNote Series 251, Cambridge University Press, 1998, 535–545. ISBN 0-521-63298-6 (This is the journalpublication of Buchberger’s thesis.)

• Buchberger, Bruno; Kauers, Manuel (2010). “Gröbner Bases”. Scholarpedia (5): 7763. doi:10.4249/scholarpedia.7763.

• Ralf Fröberg (1997). An Introduction to Gröbner Bases. Wiley & Sons. ISBN 0-471-97442-0.

• Sturmfels, Bernd (November 2005), “What is . . . a Gröbner Basis?" (PDF), Notices of the American Mathe-matical Society 52 (10): 1199–1200, a brief introduction.

• A. I. Shirshov (1999). “Certain algorithmic problems for Lie algebras” (PDF). ACM SIGSAM Bulletin 33(2): 3–6. doi:10.1145/334714.334715. (translated from Sibirsk. Mat. Zh. Siberian Mathematics Journal, 3(1962), 292–296)

• M. Aschenbrenner and C. Hillar, Finite generation of symmetric ideals, Trans. Amer. Math. Soc. 359 (2007),5171–5192 (on infinite dimensional Gröbner bases for polynomial rings in infinitely many indeterminates).

14.9 External links

• Faugère’s own implementation of his F4 algorithm

• Hazewinkel, Michiel, ed. (2001), “Gröbner basis”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

14.9. EXTERNAL LINKS 53

• B. Buchberger, Groebner Bases: A Short Introduction for Systems Theorists in Proceedings of EUROCAST2001.

• Buchberger, B. and Zapletal, A. Gröbner Bases Bibliography.

• Comparative Timings Page for Gröbner Bases Software

• Prof. Bruno Buchberger Bruno Buchberger

• Weisstein, Eric W., “Gröbner Basis”, MathWorld.

• Gröbner basis introduction on Scholarpedia

Chapter 15

Haboush’s theorem

InmathematicsHaboush’s theorem, often still referred to as theMumford conjecture, states that for any semisimplealgebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V thatis fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that

F(v) ≠ 0.

The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of V,and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p. WhenK has characteristic0 this was well known; in fact Weyl’s theorem on the complete reducibility of the representations of G implies thatF can even be taken to be linear. Mumford’s conjecture about the extension to prime characteristic p was proved byW. J. Haboush (1975), about a decade after the problem had been posed by David Mumford, in the introduction tothe first edition of his book Geometric Invariant Theory.

15.1 Applications

Haboush’s theorem can be used to generalize results of geometric invariant theory from characteristic 0, where theywere already known, to characteristic p>0. In particular Nagata’s earlier results together with Haboush’s theoremshow that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixedsubalgebra is also finitely generated.Haboush’s theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, thendisjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and 1on Y).C.S. Seshadri (1977) extended Haboush’s theorem to reductive groups over schemes.It follows from the work of Nagata (1963), Haboush, and Popov that the following conditions are equivalent for anaffine algebraic group G over a field K:

• G is reductive (its unipotent radical is trivial).

• For any non-zero invariant vector in a rational representation of G, there is an invariant homogeneous polyno-mial that does not vanish on it.

• For any finitely generatedK algebra on whichG act rationally, the algebra of fixed elements is finitely generated.

15.2 Proof

The theorem is proved in several steps as follows:

• We can assume that the group is defined over an algebraically closed field K of characteristic p>0.

54

15.3. REFERENCES 55

• Finite groups are easy to deal with as one can just take a product over all elements, so one can reduce to the caseof connected reductive groups (as the connected component has finite index). By taking a central extensionwhich is harmless one can also assume the group G is simply connected.

• Let A(G) be the coordinate ring of G. This is a representation of G with G acting by left translations. Pick anelement v′ of the dual of V that has value 1 on the invariant vector v. The map V to A(G) by sending w∈V tothe element a∈A(G) with a(g) = v′(g(w)). This sends v to 1∈A(G), so we can assume that V⊂A(G) and v=1.

• The structure of the representation A(G) is given as follows. Pick a maximal torus T of G, and let it act onA(G) by right translations (so that it commutes with the action of G). Then A(G) splits as a sum over charactersλ of T of the subrepresentations A(G)λ of elements transforming according to λ. So we can assume that V iscontained in the T-invariant subspace A(G)λ of A(G).

• The representation A(G)λ is an increasing union of subrepresentations of the form Eλ₊nᵨ⊗Enᵨ, where ρ is theWeyl vector for a choice of simple roots of T, n is a positive integer, and Eμ is the space of sections of the linebundle over G/B corresponding to a character μ of T, where B is a Borel subgroup containing T.

• If n is sufficiently large then Enᵨ has dimension (n+1)N whereN is the number of positive roots. This is becausein characteristic 0 the corresponding module has this dimension by the Weyl character formula, and for n largeenough that the line bundle over G/B is very ample, Enᵨ has the same dimension as in characteristic 0.

• If q=pr for a positive integer r, and n=q−1, thenEnᵨ contains the Steinberg representation ofG(Fq) of dimensionqN . (Here Fq ⊂ K is the finite field of order q.) The Steinberg representation is an irreducible representation ofG(Fq) and therefore of G(K), and for r large enough it has the same dimension as Enᵨ, so there are infinitelymany values of n such that Enᵨ is irreducible.

• If Enᵨ is irreducible it is isomorphic to its dual, so Enᵨ⊗Enᵨ is isomorphic to End(Enᵨ). Therefore the T-invariant subspace A(G)λ of A(G) is an increasing union of subrepresentations of the form End(E) for repre-sentations E (of the form E₍q₋₁₎ᵨ)). However for representations of the form End(E) an invariant polynomialthat separates 0 and 1 is given by the determinant. This completes the sketch of the proof of Haboush’s theorem.

15.3 References• Demazure, Michel (1976), “Démonstration de la conjecture de Mumford (d'après W. Haboush)", SéminaireBourbaki (1974/1975: Exposés Nos. 453-−470), Lecture Notes in Math. 514, Berlin: Springer, pp. 138–144,doi:10.1007/BFb0080063, ISBN 978-3-540-07686-5, MR 0444786

• Haboush, W. J. (1975), “Reductive groups are geometrically reductive”, Ann. Of Math. (The Annals ofMathematics, Vol. 102, No. 1) 102 (1): 67–83, doi:10.2307/1970974, JSTOR 1970974

• Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematikund ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994.xiv+292 pp. MR 1304906 ISBN 3-540-56963-4

• Nagata, Masayoshi (1963), “Invariants of a group in an affine ring”, Journal of Mathematics of Kyoto University3: 369–377, ISSN 0023-608X, MR 0179268

• M. Nagata, T. Miyata, “Note on semi-reductive groups” J. Math. Kyoto Univ., 3 (1964) pp. 379–382

• Popov, V.L. (2001), “Mumford hypothesis”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• C.S. Seshadri, “Geometric reductivity over arbitrary base” Adv. Math., 26 (1977) pp. 225–274

Chapter 16

Hall algebra

For the more general Hall algebra of a category, see Ringel–Hall algebra.

In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes offinite abelian p-groups. It was first discussed by E. Steinitz (1901) but forgotten until it was rediscovered by PhilipHall (1959), both of whom published no more than brief summaries of their work. The Hall polynomials are thestructure constants of theHall algebra. The Hall algebra plays an important role in the theory of Kashiwara–Lusztig'scanonical bases in quantum groups. Ringel (1990) generalized Hall algebras to more general categories, such as thecategory of representations of a quiver.

16.1 Construction

A finite abelian p-groupM is a direct sum of cyclic p-power components Cpλi , where λ = (λ1, λ2, . . .) is a partitionof n called the type of M. Let gλµ,ν(p) be the number of subgroups N of M such that N has type ν and the quotientM/N has type µ . Hall proved that the functions g are polynomial functions of p with integer coefficients. Thus wemay replace p with an indeterminate q, which results in the Hall polynomials

gλµ,ν(q) ∈ Z[q].

Hall next constructs an associative ring H over Z[q] , now called the Hall algebra. This ring has a basis consistingof the symbols uλ and the structure constants of the multiplication in this basis are given by the Hall polynomials:

uµuν =∑λ

gλµ,ν(q)uλ.

It turns out that H is a commutative ring, freely generated by the elements u1n corresponding to the elementaryp-groups. The linear map from H to the algebra of symmetric functions defined on the generators by the formula

u1n 7→ q−n(n−1)/2en

(where en is the nth elementary symmetric function) uniquely extends to a ring homomorphism and the images ofthe basis elements uλ may be interpreted via the Hall–Littlewood symmetric functions. Specializing q to 1, thesesymmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.

16.2 References• Hall, Philip (1959), “The algebra of partitions”, Proceedings of the 4th Canadian mathematical congress, Banff,pp. 147–159

56

16.2. REFERENCES 57

• George Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4 (1991),no. 2, 365–421.

• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nded.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144

• Ringel, Claus Michael (1990), “Hall algebras and quantum groups”, Inventiones Mathematicae 101 (3): 583–591, doi:10.1007/BF01231516, MR 1062796

• Schiffmann, O (2006), Lectures on Hall algebras, arXiv:math/0611617

• Steinitz, E. (1901), “Zur Theorie derAbel’schenGruppen”, Jahresbericht der DeutschenMathematiker-Vereinigung9: 80–85

Chapter 17

Hermite reciprocity

In mathematics,Hermite’s law of reciprocity, introduced by Hermite (1854), states that the degreem covariants of abinary form of degree n correspond to the degree n covariants of a binary form of degreem. In terms of representationtheory it states that the representations Sm Sn C2 and Sn Sm C2 of GL2 are isomorphic.

17.1 References• Hermite, Charles (1854), “Sur la theorie des fonctions homogenes à deux indéterminées”, Cambridge andDublin Mathematical Journal 9: 172–217

58

Chapter 18

Hilbert’s basis theorem

In mathematics, specifically commutative algebra, Hilbert’s basis theorem says that a polynomial ring over aNoetherian ring is Noetherian.

18.1 Statement

If R a ring, let R[X] denote the ring of polynomials in the indeterminateX over R . Hilbert proved that if R is “nottoo large”, in the sense that if R is Noetherian, the same must be true for R[X] . Formally,

Hilbert’s Basis Theorem. If R is a Noetherian ring, then R[X] is a Noetherian ring.

Corollary. If R is a Noetherian ring, then R[X1, . . . , Xn] is a Noetherian ring.

This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the setof common roots of finitely many polynomial equations. Hilbert (1890) proved the theorem (for the special case ofpolynomial rings over a field) in the course of his proof of finite generation of rings of invariants.Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give analgorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. Onecan determine basis polynomials using the method of Gröbner bases.

18.2 ProofTheorem. If R is a left (resp. right) Noetherian ring, then the polynomial ring R[X] is also a left (resp.right) Noetherian ring.

Remark. We will give two proofs, in both only the “left” case is considered, the proof for the right case is similar.

18.2.1 First Proof

Suppose a ⊆ R[X] were a non-finitely generated left-ideal. Then by recursion (using the axiom of dependent choice)there is a sequence f0, f1, . . . of polynomials such that if bn is the left ideal generated by f0, . . . , fn−1 then fn ina \ bn is of minimal degree. It is clear that deg(f0), deg(f1), . . . is a non-decreasing sequence of naturals. Let anbe the leading coefficient of fn and let b be the left ideal in R generated by a0, a1, . . . . Since R is Noetherian thechain of ideals (a0) ⊂ (a0, a1) ⊂ (a0, a1, a2) . . . must terminate. Thus b = (a0, . . . , aN−1) for some integer N .So in particular,

aN =∑i<N

uiai, ui ∈ R.

59

60 CHAPTER 18. HILBERT’S BASIS THEOREM

Now consider

g =∑i<N

uiXdeg(fN )−deg(fi)fi,

whose leading term is equal to that of fN ; moreover, g ∈ bN . However, fN /∈ bN , whichmeans that fN−g ∈ a\bNhas degree less than fN , contradicting the minimality.

18.2.2 Second Proof

Let a ⊆ R[X] be a left-ideal. Let b be the set of leading coefficients of members of a . This is obviously a left-idealoverR , and so is finitely generated by the leading coefficients of finitely many members of a ; say f0, . . . , fN−1 . Letd be the maximum of the set deg(f0), . . . , deg(fN−1) , and let bk be the set of leading coefficients of membersof a , whose degree is ≤ k . As before, the bk are left-ideals over R , and so are finitely generated by the leadingcoefficients of finitely many members of a , say

f(k)0 , · · · , f (k)

N(k)−1,

with degrees ≤ k . Now let a∗ ⊆ R[X] be the left-ideal generated by

fi, f

(k)j : i < N, j < N (k), k < d

.

We have a∗ ⊆ a and claim also a ⊆ a∗ . Suppose for the sake of contradiction this is not so. Then let h ∈ a \ a∗ beof minimal degree, and denote its leading coefficient by a .

Case 1: deg(h) ≥ d . Regardless of this condition, we have a ∈ b , so is a left-linear combination

a =∑j

ujaj

fj

h0 ≜∑j

ujXdeg(h)−deg(fj)fj ,

which has the same leading term as h ; moreover h0 ∈ a∗ while h /∈ a∗ . Therefore h − h0 ∈ a \ a∗and deg(h− h′0) < deg(h) , which contradicts minimality.

Case 2: deg(h) = k < d . Then a ∈ bk so is a left-linear combination

a =∑j

uja(k)j

f(k)j

h0 ≜∑j

ujXdeg(h)−deg(f(k)

j )f(k)j ,

we yield a similar contradiction as in Case 1.

Thus our claim holds, and a = a∗ which is finitely generated.Note that the only reason we had to split into two cases was to ensure that the powers of X multiplying the factors,were non-negative in the constructions.

18.3. APPLICATIONS 61

18.3 Applications

Let R be a Noetherian commutative ring. Hilbert’s basis theorem has some immediate corollaries.

1. By induction we see that R[X0, . . . , Xn−1] will also be Noetherian.

2. Since any affine variety over Rn (i.e. a locus-set of a collection of polynomials) may be written as the locus ofan ideal a ⊂ R[X0, . . . , Xn−1] and further as the locus of its generators, it follows that every affine variety isthe locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces.

3. If A is a finitely-generatedR -algebra, then we know that A ≃ R[X0, . . . , Xn−1]/a , where a is an ideal. Thebasis theorem implies that a must be finitely generated, say a = (p0, . . . , pN−1) , i.e. A is finitely presented.

18.4 Mizar System

The Mizar project has completely formalized and automatically checked a proof of Hilbert’s basis theorem in theHILBASIS file.

18.5 References• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.

• Hilbert, David (1890), “Ueber die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831

Chapter 19

Hilbert’s fourteenth problem

In mathematics, Hilbert’s fourteenth problem, that is, number 14 of Hilbert’s problems proposed in 1900, askswhether certain algebras are finitely generated.The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables,

k(x1, ..., xn ) over k.

Consider now the k-algebra R defined as the intersection

R := K ∩ k[x1, . . . , xn] .

Hilbert conjectured that all such algebras are finitely generated over k.After some results were obtained confirming Hilbert’s conjecture in special cases and for certain classes of rings (inparticular the conjecture was proved unconditionally for n = 1 and n = 2 by Zariski in 1954) then in 1959 MasayoshiNagata found a counterexample to Hilbert’s conjecture. The counterexample of Nagata is a suitably constructed ringof invariants for the action of a linear algebraic group.

19.1 History

The problem originally arose in algebraic invariant theory. Here the ring R is given as a (suitably defined) ring ofpolynomial invariants of a linear algebraic group over a field k acting algebraically on a polynomial ring k[x1, ...,xn] (or more generally, on a finitely generated algebra defined over a field). In this situation the field K is the fieldof rational functions (quotients of polynomials) in the variables xi which are invariant under the given action of thealgebraic group, the ring R is the ring of polynomials which are invariant under the action. A classical example innineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and also Hilbert)of invariants of binary forms in two variables with the natural action of the special linear group SL2(k) on it. Hilberthimself proved the finite generation of invariant rings in the case of the field of complex numbers for some classicalsemi-simple Lie groups (in particular the general linear group over the complex numbers) and specific linear actionson polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group. This finitenessresult was later extended by HermannWeyl to the class of all semi-simple Lie-groups. A major ingredient in Hilbert’sproof is the Hilbert basis theorem applied to the ideal inside the polynomial ring generated by the invariants.

19.2 Zariski’s formulation

Zariski's formulation of Hilbert’s fourteenth problem asks whether, for a quasi-affine algebraic variety X over a fieldk, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k.Zariski’s formulation was shown[1] to be equivalent to the original problem, for X normal.

62

19.3. NAGATA’S COUNTEREXAMPLE 63

19.3 Nagata’s counterexample

Nagata (1958) gave the following counterexample to Hilbert’s problem. The field k is a field containing 48 elementsa₁i, ...,a₁₆i, for i=1, 2, 3 that are algebraically independent over the prime field. The ring R is the polynomial ringk[x1,...,x16, t1,...,t16] in 32 variables. The vector space V is a 13-dimensional vector space over k consisting of allvectors (b1,...,b16) in k16 orthogonal to each of the three vectors (a₁i, ...,a₁₆i) for i=1, 2, 3. The vector space V is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on R by fixing all elementstj and taking xj to xj + bjtj. Then the ring of elements of R invariant under the action of the group V is not a finitelygenerated k-algebra.Several authors have reduced the sizes of the group and the vector space in Nagata’s example. For example, Totaro(2008) showed that over any field there is an action of the sum G3a of three copies of the additive group on k18 whose ring of invariants is not finitely generated.

19.4 References• Nagata, Masayoshi (1960), “On the fourteenth problem of Hilbert”, Proc. Internat. Congress Math. 1958,Cambridge University Press, pp. 459–462, MR 0116056

• Nagata, Masayoshi (1965), Lectures on the fourteenth problem of Hilbert (PDF), Tata Institute of FundamentalResearch Lectures on Mathematics 31, Bombay: Tata Institute of Fundamental Research, MR 0215828

• Totaro, Burt (2008), “Hilbert’s 14th problem over finite fields and a conjecture on the cone of curves”, Compo-sitio Mathematica 144 (5): 1176–1198, doi:10.1112/S0010437X08003667, ISSN 0010-437X, MR 2457523

• O. Zariski, Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert, Bulletin des SciencesMathematiques 78 (1954), pp. 155–168.

[1] Winkelmann, Jörg (2003), “Invariant rings and quasiaffine quotients”, Math. Z. 244 (1): 163–174, doi:10.1007/s00209-002-0484-9.

Chapter 20

Hilbert’s syzygy theorem

In mathematics,Hilbert’s syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890) inconnection with the syzygy (relation) problem of invariant theory. Roughly speaking, starting with relations betweenpolynomial invariants, then relations between the relations, and so on, it explains how far one has to go to reach aclarified situation. It is now considered to be an early result of homological algebra, and through the depth concept,to be a measure of the non-singularity of affine space.

20.1 Formal statement

In modern language, the theorem may be stated as follows. Let k be a field and M a finitely generated module overthe polynomial ring

k[x1, . . . , xn].

Hilbert’s syzygy theorem then states that there exists a free resolution of M of length at most n.

20.2 See also• Quillen–Suslin theorem

• Hilbert polynomial

20.3 References• David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathemat-ics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 MR1322960

• Hazewinkel, Michiel, ed. (2001), “Hilbert theorem”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

64

Chapter 21

Hilbert–Mumford criterion

In mathematics, the Hilbert–Mumford criterion, introduced by David Hilbert and David Mumford, characterizesthe semistable and stable points of a group action on a vector space in terms of eigenvalues of 1-parameter subgroups(Dieudonné & Carrell 1970, 1971, p.58).

21.1 References• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525

• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant Theory, Old and New, Boston, MA: Academic Press,ISBN 978-0-12-215540-6, MR 0279102

65

Chapter 22

Hodge bundle

In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, whereit provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory ofmodular forms on reductive algebraic groups[1] and string theory.[2]

22.1 Definitions

Let Mg be the moduli space of algebraic curves of genus g curves over some scheme. The Hodge bundle Λg is avector bundle on Mg whose fiber at a point C in Mg is the space of holomorphic differentials on the curve C. Todefine the Hodge bundle, let π : Cg → Mg be the universal algebraic curve of genus g and let ωg be its relativedualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.[3]

Λg = π∗ωg.

22.2 See also• ELSV formula

22.3 Notes[1] van der Geer 2008, §13

[2] Liu 2006, §5

[3] Harris & Morrison 1998, p. 155

22.4 References• van der Geer, Gerard (2008), “Siegel modular forms and their applications”, in Ranestad, Kristian, The 1-2-3 of modular forms, Universitext, Berlin: Springer-Verlag, pp. 181–245, doi:10.1007/978-3-540-74119-0,ISBN 978-3-540-74117-6, MR 2409679

• Harris, Joe; Morrison, Ian (1998), Moduli of curves, Graduate Texts in Mathematics 187, Springer-Verlag,ISBN 978-0-387-98429-2, MR 1631825

• Liu, Kefeng (2006), “Localization and conjectures from string duality”, in Ge, Mo-Lin; Zhang, Weiping,Differential geometry and physics, Nankai Tracts in Mathematics 10, World Scientific, pp. 63–105, ISBN978-981-270-377-4, MR 2322389

66

Chapter 23

Invariant estimator

In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the proper-ties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should havecertain intuitively appealing qualities. Strictly speaking, “invariant” would mean that the estimates themselves areunchanged when both the measurements and the parameters are transformed in a compatible way, but the meaninghas been extended to allow the estimates to change in appropriate ways with such transformations.[1] The term equiv-ariant estimator is used in formal mathematical contexts that include a precise description of the relation of theway the estimator changes in response to changes to the dataset and parameterisation: this corresponds to the use of"equivariance" in more general mathematics.

23.1 General setting

23.1.1 Background

In statistical inference, there are several approaches to estimation theory that can be used to decide immediately whatestimators should be used according to those approaches. For example, ideas from Bayesian inference would leaddirectly to Bayesian estimators. Similarly, the theory of classical statistical inference can sometimes lead to strongconclusions about what estimator should be used. However, the usefulness of these theories depends on having a fullyprescribed statistical model and may also depend on having a relevant loss function to determine the estimator. Thusa Bayesian analysis might be undertaken, leading to a posterior distribution for relevant parameters, but the use of aspecific utility or loss function may be unclear. Ideas of invariance can then be applied to the task of summarising theposterior distribution. In other cases, statistical analyses are undertaken without a fully defined statistical model orthe classical theory of statistical inference cannot be readily applied because the family of models being consideredare not amenable to such treatment. In addition to these cases where general theory does not prescribe an estimator,the concept of invariance of an estimator can be applied when seeking estimators of alternative forms, either for thesake of simplicity of application of the estimator or so that the estimator is robust.The concept of invariance is sometimes used on its own as a way of choosing between estimators, but this is notnecessarily definitive. For example, a requirement of invariance may be incompatible with the requirement thatthe estimator be mean-unbiased; on the other hand, the criterion of median-unbiasedness is defined in terms of theestimator’s sampling distribution and so is invariant under many transformations.One use of the concept of invariance is where a class or family of estimators is proposed and a particular formulationmust be selected amongst these. One procedure is to impose relevant invariance properties and then to find theformulation within this class that has the best properties, leading to what is called the optimal invariant estimator.

23.1.2 Some classes of invariant estimators

There are several types of transformations that are usefully considered when dealing with invariant estimators. Eachgives rise to a class of estimators which are invariant to those particular types of transformation.

• Shift invariance: Notionally, estimates of a location parameter should be invariant to simple shifts of the data

67

68 CHAPTER 23. INVARIANT ESTIMATOR

values. If all data values are increased by a given amount, the estimate should change by the same amount.When considering estimation using a weighted average, this invariance requirement immediately implies thatthe weights should sum to one. While the same result is often derived from a requirement for unbiasedness, theuse of “invariance” does not require that a mean value exists and makes no use of any probability distributionat all.

• Scale invariance: Note that this topic about the invariance of the estimator scale parameter not to be confusedwith the more general scale invariance about the behavior of systems under aggregate properties (in physics).

• Parameter-transformation invariance: Here, the transformation applies to the parameters alone. The concepthere is that essentially the same inference should be made from data and a model involving a parameter θ aswould be made from the same data if the model used a parameter φ, where φ is a one-to-one transformation ofθ, φ=h(θ). According to this type of invariance, results from transformation-invariant estimators should alsobe related by φ=h(θ). Maximum likelihood estimators have this property. Though the asymptotic propertiesof the estimator might be invariant, the small sample properties can be different, and a specific distributionneeds to be derived.[2]

• Permutation invariance: Where a set of data values can be represented by a statistical model that they areoutcomes from independent and identically distributed random variables, it is reasonable to impose the re-quirement that any estimator of any property of the common distribution should be permutation-invariant:specifically that the estimator, considered as a function of the set of data-values, should not change if items ofdata are swapped within the dataset.

The combination of permutation invariance and location invariance for estimating a location parameter from anindependent and identically distributed dataset using a weighted average implies that the weights should be identicaland sum to one. Of course, estimators other than a weighted average may be preferable.

23.1.3 Optimal invariant estimators

Under this setting, we are given a set of measurements x which contains information about an unknown parameter θ. The measurements x are modelled as a vector random variable having a probability density function f(x|θ) whichdepends on a parameter vector θ .The problem is to estimate θ given x . The estimate, denoted by a , is a function of the measurements and belongsto a set A . The quality of the result is defined by a loss function L = L(a, θ) which determines a risk functionR = R(a, θ) = E[L(a, θ)|θ] . The sets of possible values of x , θ , and a are denoted by X , Θ , and A ,respectively.

23.1.4 In classification

In statistical classification, the rule which assigns a class to a new data-item can be consider to be a special type ofestimator. A number of invariance-type considerations can be brought to bear in formulating prior knowledge forpattern recognition.

23.2 Mathematical setting

23.2.1 Definition

An invariant estimator is an estimator which obeys the following two rules:

1. Principle of Rational Invariance: The action taken in a decision problem should not depend on transformationon the measurement used

2. Invariance Principle: If two decision problems have the same formal structure (in terms ofX ,Θ , f(x|θ) andL ), then the same decision rule should be used in each problem.

23.2. MATHEMATICAL SETTING 69

To define an invariant or equivariant estimator formally, some definitions related to groups of transformations areneeded first. LetX denote the set of possible data-samples. A group of transformations ofX , to be denoted by G ,is a set of (measurable) 1:1 and onto transformations ofX into itself, which satisfies the following conditions:

1. If g1 ∈ G and g2 ∈ G then g1g2 ∈ G

2. If g ∈ G then g−1 ∈ G , where g−1(g(x)) = x . (That is, each transformation has an inverse within thegroup.)

3. e ∈ G (i.e. there is an identity transformation e(x) = x )

Datasets x1 and x2 inX are equivalent if x1 = g(x2) for some g ∈ G . All the equivalent points form an equivalenceclass. Such an equivalence class is called an orbit (inX ). The x0 orbit,X(x0) , is the setX(x0) = g(x0) : g ∈ G. If X consists of a single orbit then g is said to be transitive.A family of densities F is said to be invariant under the groupG if, for every g ∈ G and θ ∈ Θ there exists a uniqueθ∗ ∈ Θ such that Y = g(x) has density f(y|θ∗) . θ∗ will be denoted g(θ) .If F is invariant under the group G then the loss function L(θ, a) is said to be invariant under G if for every g ∈ Gand a ∈ A there exists an a∗ ∈ A such that L(θ, a) = L(g(θ), a∗) for all θ ∈ Θ . The transformed value a∗ will bedenoted by g(a) .In the above, G = g : g ∈ G is a group of transformations from Θ to itself and G = g : g ∈ G is a group oftransformations from A to itself.An estimation problem is invariant(equivariant) under G if there exist three groups G, G, G as defined above.For an estimation problem that is invariant underG , estimator δ(x) is an invariant estimator underG if, for all x ∈ Xand g ∈ G ,

δ(g(x)) = g(δ(x)).

23.2.2 Properties

1. The risk function of an invariant estimator, δ , is constant on orbits of Θ . Equivalently R(θ, δ) = R(g(θ), δ)for all θ ∈ Θ and g ∈ G .

2. The risk function of an invariant estimator with transitive g is constant.

For a given problem, the invariant estimator with the lowest risk is termed the “best invariant estimator”. Best invariantestimator cannot always be achieved. A special case for which it can be achieved is the case when g is transitive.

23.2.3 Example: Location parameter

Suppose θ is a location parameter if the density ofX is of the form f(x− θ) . ForΘ = A = R1 and L = L(a− θ), the problem is invariant under g = g = g = gc : gc(x) = x + c, c ∈ R . The invariant estimator in this casemust satisfy

δ(x+ c) = δ(x) + c, all for c ∈ R,

thus it is of the form δ(x) = x + K ( K ∈ R ). g is transitive on Θ so the risk does not vary with θ : that is,R(θ, δ) = R(0, δ) = E[L(X +K)|θ = 0] . The best invariant estimator is the one that brings the risk R(θ, δ) tominimum.In the case that L is the squared error δ(x) = x− E[X|θ = 0].

70 CHAPTER 23. INVARIANT ESTIMATOR

23.2.4 Pitman estimator

The estimation problem is that X = (X1, . . . , Xn) has density f(x1 − θ, . . . , xn − θ) , where θ is a parameterto be estimated, and where the loss function is L(|a − θ|) . This problem is invariant with the following (additive)transformation groups:

G = gc : gc(x) = (x1 + c, . . . , xn + c), c ∈ R1,

G = gc : gc(θ) = θ + c, c ∈ R1,G = gc : gc(a) = a+ c, c ∈ R1.The best invariant estimator δ(x) is the one that minimizes

∫∞−∞ L(δ(x)− θ)f(x1 − θ, . . . , xn − θ)dθ∫∞

−∞ f(x1 − θ, . . . , xn − θ)dθ,

and this is Pitman’s estimator (1939).For the squared error loss case, the result is

δ(x) =

∫∞−∞ θf(x1 − θ, . . . , xn − θ)dθ∫∞−∞ f(x1 − θ, . . . , xn − θ)dθ

.

If x ∼ N(θ1n, I) (i.e. a multivariate normal distribution with independent, unit-variance components) then

δpitman = δML =

∑xin

.

If x ∼ C(θ1n, Iσ2) (independent components having a Cauchy distribution with scale parameter σ) then δpitman =

δML ,. However the result is

δpitman =

n∑k=1

xk

[Rewk∑nm=1Rewk

], n > 1,

with

wk =∏j =k

[1

(xk − xj)2 + 4σ2

] [1− 2σ

(xk − xj)i

].

23.3 References[1] see section 5.2.1 in Gourieroux, C. and Monfort, A. (1995). Statistics and econometric models, volume 1. Cambridge

University Press.

[2] Gourrieroux and Monfort (1995)

• Berger, James O. (1985). Statistical decision theory and Bayesian Analysis (2nd ed.). New York: Springer-Verlag. ISBN 0-387-96098-8. MR 0804611.

• Freue, Gabriela V. Cohen (2007) “The Pitman estimator of the Cauchy location parameter”, Journal of Sta-tistical Planning and Inference, 137, 1900–1913 doi:10.1016/j.jspi.2006.05.002

• Pitman, E.J.G. (1939) “The estimation of the location and scale parameters of a continuous population of anygiven form”, Biometrika, 30 (3/4), 391–421. JSTOR 2332656

• Pitman, E.J.G. (1939) “Tests of Hypotheses Concerning Location and Scale Parameters”, Biometrika, 31 (1/2),200–215. JSTOR 2334983

Chapter 24

Invariant of a binary form

In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary formin two variables x and y that remains invariant under the special linear group acting on the variables x and y.

24.1 Terminology

Main article: Glossary of invariant theory

A binary form (of degree n) is a homogeneous polynomial Σni=0 (ni)an₋ixn−iyi = anxn + (n1)an₋₁xn−1y + ... + a0yn. The group SL2(C) acts on these forms by taking x to ax + by and y to cx + dy. This inducesan action on the space spanned by a0, ..., an and on the polynomials in these variables. An invariant is a polynomialin these n + 1 variables a0, ..., an that is invariant under this action. More generally a covariant is a polynomial in a0,..., an, x, y that is invariant, so an invariant is a special case of a covariant where the variables x and y do not occur.More generally still, a simultaneous invariant is a polynomial in the coefficients of several different forms in x andy.In terms of representation theory, given any representation V of the group SL2(C) one can ask for the ring of invariantpolynomials on V. Invariants of a binary form of degree n correspond to taking V to be the (n + 1)-dimensionalirreducible representation, and covariants correspond to taking V to be the sum of the irreducible representations ofdimensions 2 and n + 1.The invariants of a binary form form a graded algebra, and Gordan (1868) proved that this algebra is finitely generatedif the base field is the complex numbers.Forms of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10 are sometimes called quadrics, cubic, quartics, quintics, sextics, septics orseptimics, octics or octavics, nonics, and decics or decimics. “Quantic” is an old name for a form of arbitrary degree.Forms in 1, 2, 3, 4, ... variables are called unary, binary, ternary, quaternary, ... forms.

24.2 Examples

A form f is itself a covariant of degree 1 and order n.The discriminant of a form is an invariant.The resultant of two forms is a simultaneous invariant of them.The Hessian covariant of a form Hilbert (1993, p.88) is the determinant of the Hessian matrix

71

72 CHAPTER 24. INVARIANT OF A BINARY FORM

H(f) =

∂2f∂x2

∂2f∂x ∂y

∂2f∂y ∂x

∂2f∂y2

.It is a covariant of order 2n− 4 and degree 2.The catalecticant is an invariant of degree n/2+1 of a binary form of even degree n.The canonizant is a covariant of degree and order (n+1)/2 of a binary form of odd degree n.The Jacobian

det

∂f∂x

∂f∂y

∂g∂x

∂g∂y

is a simultaneous invariant of two forms f, g.

24.3 The ring of invariants

The structure of the ring of invariants has been worked out for small degrees. Sylvester & Franklin (1879) gave tablesof the numbers of generators of invariants and covariants for forms of degree up to 10, though the tables have a fewminor errors for large degrees, mostly where a few invariants or covariants are omitted.

24.3.1 Covariants of a binary linear form

For linear forms ax + by the only invariants are constants. The algebra of covariants is generated by the form itselfof degree 1 and order 1.

24.3.2 Covariants of a binary quadric

The algebra of invariants of the quadratic form ax2 + 2bxy + cy2 is a polynomial algebra in 1 variable generated bythe discriminant b2 − ac of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated bythe discriminant together with the form f itself (of degree 1 and order 2). (Schur 1968, II.8) (Hilbert 1993, XVI,XX)

24.3.3 Covariants of a binary cubic

The algebra of invariants of the cubic form ax3 + 3bx2y + 3cxy2 + dy3 is a polynomial algebra in 1 variable generatedby the discriminant D = 3b2c2 + 6abcd − 4b3d − 4c3a − a2d2 of degree 4. The algebra of covariants is generated bythe discriminant, the form itself (degree 1, order 3), the Hessian H (degree 2, order 2) and a covariant T of degree 3and order 3. They are related by the syzygy 4h3=Df2-T2 of degree 6 and order 6. (Schur 1968, II.8) (Hilbert 1993,XVII, XX)

24.3.4 Covariants of a binary quartic

The algebra of invariants of a quartic form is generated by invariants i, j of degrees 2, 3. This ring is naturallyisomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series E4

and E6. The algebra of covariants is generated by these two invariants together with the form f of degree 1 and order4, the Hessian H of degree 2 and order 4, and a covariant T of degree 3 and order 6. They are related by a syzygyjf3−Hf2i+4H3+T2=0 of degree 6 and order 12. (Schur 1968, II.8) (Hilbert 1993, XVIII, XXII)

24.3. THE RING OF INVARIANTS 73

24.3.5 Covariants of a binary quintic

The algebra of invariants of a quintic formwas found by Sylvester and is generated by invariants of degree 4, 8, 12, 18.The generators of degrees 4, 8, 12 generate a polynomial ring, which contains the square of Hermite’s skew invariantof degree 18. The invariants are rather complicated to write out explicitly: Sylvester showed that the generators ofdegrees 4, 8, 12, 18 have 12, 59, 228, and 848 terms often with very large coefficients. (Schur 1968, II.9) (Hilbert1993, XVIII) The ring of covariants is generated by 23 covariants, one of which is the canonizant of degree 3 andorder 3.

24.3.6 Covariants of a binary sextic

The algebra of invariants of a sextic form is generated by invariants of degree 2, 4, 6, 10, 15. The generators ofdegrees 2, 4, 6, 10 generate a polynomial ring, which contains the square of the generator of degree 15. (Schur 1968,II.9) The ring of covariants is generated by 26 covariants. The ring of invariants is closely related to the moduli spaceof curves of genus 2, because such a curve can be represented as a double cover of the projective line branched at 6points, and the 6 points can be taken as the roots of a binary sextic.

24.3.7 Covariants of a binary septic

The ring of invariants of binary septics is anomalous and has caused several published errors. Cayley claimed incor-rectly that the ring of invariants is not finitely generated. Sylvester & Franklin (1879) gave lower bounds of 26 and124 for the number of generators of the ring of invariants and the ring of covariants and observed that an unproved“fundamental postulate” would imply that equality holds. However von Gall (1888) showed that Sylvester’s numbersare not equal to the numbers of generators, which are 30 for the ring of invariants and at least 130 for the ring ofcovariants, so Sylvester’s fundamental postulate is wrong. von Gall (1888) and Dixmier & Lazard (1986) showedthat the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6of degree 12, 4 of degree 14, 2 of degree 16, 9 of degree 18, and one of each of the degrees 20, 22, 26, 30. Cröni(2002) gives 147 generators for the ring of covariants.

24.3.8 Covariants of a binary octavic

Sylvester & Franklin (1879) showed that the ring of invariants of a degree 8 form is generated by 9 invariants ofdegrees 2, 3, 4, 5, 6, 7, 8, 9, 10, and the ring of covariants is generated by 69 covariants. August von Gall (von Gall(1880)) and Shioda (1967) confirmed the generators for the ring of invariants and showed that the ideal of relationsbetween them is generated by elements of degrees 16, 17, 18, 19, 20.

24.3.9 Covariants of a binary nonic

Brouwer & Popoviciu (2010a) showed that the algebra of invariants of a degree 9 form is generated by 92 invariants.

24.3.10 Covariants of a binary decimic

Sylvester stated that the ring of invariants of binary decics is generated by 104 invariants the ring of covariants by 475covariants; his list is to be correct for degrees up to 16 but wrong for higher degrees. Brouwer & Popoviciu (2010b)showed that the algebra of invariants of a degree 10 form is generated by 106 invariants

24.3.11 Covariants of a binary undecimic

The ring of invariants of binary forms of degree 11 is complicated and has not yet been described explicitly.

74 CHAPTER 24. INVARIANT OF A BINARY FORM

24.3.12 Covariants of a binary duodecimic

For forms of degree 12 Sylvester (1881) found that in degrees up to 14 there are 109 basic invariants. There are atleast 4 more in higher degrees. The number of basic covariants is at least 989.The number of generators for invariants and covariants of binary forms can be found in (sequence A036983 in OEIS)and (sequence A036984 in OEIS), respectively.

24.4 Invariants of several binary forms

The covariants of a binary form are essentially the same as joint invariants of a binary form and a binary linear form.More generally, on can ask for the joint invariants (and covariants) of any collection of binary forms. Some casesthat have been studied are listed below.

24.4.1 Covariants of two linear forms

There are 1 basic invariant and 3 basic covariants.

24.4.2 Covariants of a linear form and a quadratic

There are 2 basic invariants and 5 basic covariants.

24.4.3 Covariants of a linear form and a cubic

There are 4 basic invariants (essentially the covariants of a cubic) and 13 basic covariants.

24.4.4 Covariants of a linear form and a quartic

There are 5 basic invariants (essentially the basic covariants of a quartic) and 20 basic covariants.

24.4.5 Covariants of a linear form and a quintic

There are 23 basic invariants (essentially the basic covariants of a quintic) and 94 basic covariants.

24.4.6 Covariants of a linear form and a quantic

24.4.7 Covariants of several linear forms

The ring of invariants of n linear forms is generated by n(n–1)/2 invariants of degree 2. The ring of covariants of nlinear forms is essentially the same as the ring of invariants of n+1 linear forms.

24.4.8 Covariants of two quadratics

There are 3 basic invariants and 6 basic covariants.

24.4.9 Covariants of two quadratics and a linear form

24.4.10 Covariants of several linear and quadratic forms

The ring of invariants of a sum of m linear forms and n quadratic forms is generated by m(m–1)/2 + n(n+1)/2generators in degree 2, nm (m+1)/2 + n(n–1)(n–2)/6 in degree 3, and m(m+1)n(n –1)/4 in degree 4.

24.5. SEE ALSO 75

For the number of generators of the ring of covariants, change m to m+1.

24.4.11 Covariants of a quadratic and a cubic

There are 5 basic invariants and 15 basic covariants

24.4.12 Covariants of a quadratic and a quartic

There are 6 basic invariants and 18 basic covariants

24.4.13 Covariants of a quadratic and a quintic

There are 29 basic invariants and 92 basic covariants

24.4.14 Covariants of a cubic and a quartic

There are 20 basic invariants and 63 basic covariants

24.4.15 Covariants of two quartics

There are 8 basic invariants (3 of degree 2, 4 of degree 3, and 1 of degree 4) and 28 basic covariants. (Gordan gave30 covariants, but Sylvester showed that two of these are reducible.)

24.4.16 Covariants of many cubics or quartics

The numbers of generators of invariants or covariants were given by Young (1899).

24.5 See also

• Ternary cubic

• Ternary quartic

24.6 References

• Brouwer, Andries E.; Popoviciu, Mihaela (2010a), “The invariants of the binary nonic”, Journal of SymbolicComputation 45 (6): 709–720, doi:10.1016/j.jsc.2010.03.003, ISSN 0747-7171, MR 2639312

• Brouwer, Andries E.; Popoviciu, Mihaela (2010b), “The invariants of the binary decimic”, Journal of SymbolicComputation 45 (8): 837–843, doi:10.1016/j.jsc.2010.03.002, ISSN 0747-7171, MR 2657667

• Dixmier, Jacques; Lazard, D. (1988), “Minimum number of fundamental invariants for the binary form ofdegree 7”, Journal of Symbolic Computation 6 (1): 113–115, doi:10.1016/S0747-7171(88)80026-9, ISSN0747-7171, MR 961375

• von Gall, August Freiherr (1880), “Das vollständige Formensystem einer binären Form achter Ordnung”,Mathematische Annalen 17 (1): 31–51, doi:10.1007/BF01444117, ISSN 0025-5831, MR 1510048

• vonGall, August Freiherr (1888), “Das vollständige Formensystem der binären Form 7terOrdnung”,MathematischeAnnalen 31 (3): 318–336, doi:10.1007/BF01206218, ISSN 0025-5831, MR 1510486

76 CHAPTER 24. INVARIANT OF A BINARY FORM

• Gordan, Paul (1868), “Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktionmit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist”, J. F. Math 69 (69): 323–354,doi:10.1515/crll.1868.69.323

• Hilbert, David (1993) [1897], Theory of algebraic invariants, Cambridge University Press, ISBN 978-0-521-44457-6, MR 1266168

• Kung, Joseph P. S.; Rota, Gian-Carlo (1984), “The invariant theory of binary forms”, American MathematicalSociety. Bulletin. New Series 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904, MR722856

• Schur, Issai (1968), Grunsky, Helmut, ed., Vorlesungen über Invariantentheorie, Die Grundlehren der mathe-matischen Wissenschaften 143, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04139-9, MR 0229674

• Shioda, Tetsuji (1967), “On the graded ring of invariants of binary octavics”, American Journal of Mathematics89 (4): 1022–1046, doi:10.2307/2373415, ISSN 0002-9327, JSTOR 2373415, MR 0220738

• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN978-3-211-82445-0,MR1255980

• Sylvester, J. J.; Franklin, F. (1879), “Tables of the Generating Functions and Groundforms for the BinaryQuantics of the First Ten Orders”, American Journal of Mathematics 2 (3): 223–251, doi:10.2307/2369240,ISSN 0002-9327, MR 1505222

• Sylvester, James Joseph (1881), “Tables of the Generating Functions and Groundforms of the Binary Duodec-imic, with Some General Remarks, and Tables of the Irreducible Syzygies of Certain Quantics”, AmericanJournal of Mathematics (The Johns Hopkins University Press) 4 (1): 41–61, doi:10.2307/2369149, ISSN0002-9327

24.7 External links• Brouwer, Andries E., Invariants of binary forms

Chapter 25

Invariant polynomial

In mathematics, an invariant polynomial is a polynomial P that is invariant under a group Γ acting on a vector spaceV . Therefore P is a Γ -invariant polynomial if

P (γx) = P (x)

for all γ ∈ Γ and x ∈ V .Cases of particular importance are for Γ a finite group (in the theory of Molien series, in particular), a compact group,a Lie group or algebraic group. For a basis-independent definition of 'polynomial' nothing is lost by referring to thesymmetric powers of the given linear representation of Γ.

25.1 References• This article incorporates material from Invariant polynomial on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

77

Chapter 26

Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vectorspaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicitdescription of polynomial functions that do not change, or are invariant, under the transformations from a givenlinear group. For example, if we consider the action of the special linear group SLn on the space of n by n matricesby left multiplication, then the determinant is an invariant of this action because the determinant of A X equals thedeterminant of X, when A is in SLn.

26.1 Introduction

LetG be a group, andV a finite-dimensional vector space over a field k (which in classical invariant theory was usuallyassumed to be the complex numbers). A representation of G in V is a group homomorphism π : G → GL(V ) ,which induces a group action of G on V. If k[V] is the space of polynomial functions on V, then the group action ofG on V produces an action on k[V] by the following formula:

(g · f)(x) := f(g−1x) ∀x ∈ V, g ∈ G, f ∈ k[V ].

With this action it is natural to consider the subspace of all polynomial functions which are invariant under this groupaction, in other words the set of polynomials such that g.f = f for all g in G. This space of invariant polynomials isdenoted k[V]G.First problem of invariant theory:[1] Is k[V]G a finitely generated algebra over k?For example, ifG=SLn andV=Mn to space of square matrices, and the action ofG onV is given by left multiplication,then k[V]G is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, inthis case, every invariant polynomial is a linear combination of power of the determinant polynomial. So in this case,k[V]G is finitely generated over k.If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomialrelations between the basis elements (known as the syzygies) is finitely generated over k[V].Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results wasthe main theorem on the symmetric functions that described the invariants of the symmetric group Sn acting onthe polynomial ring R[x1, …, xn] by permutations of the variables. More generally, the Chevalley–Shephard–Toddtheorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invarianttheory of finite groups emphasizes “effective” results, such as explicit bounds on the degrees of the generators. Thecase of positive characteristic, ideologically close to modular representation theory, is an area of active study, withlinks to algebraic topology.Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theoriesof quadratic forms and determinants. Another subject with strong mutual influence was projective geometry, whereinvariant theory was expected to play a major role in organizing the material. One of the highlights of this relationshipis the symbolic method. Representation theory of semisimple Lie groups has its roots in invariant theory.

78

26.2. THE NINETEENTH-CENTURY ORIGINS 79

David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in the creationof a new mathematical discipline, abstract algebra. A later paper of Hilbert (1893) dealt with the same questions inmore constructive and geometric ways, but remained virtually unknown until David Mumford brought these ideasback to life in the 1960s, in a considerably more general and modern form, in his geometric invariant theory. In largemeasure due to the influence of Mumford, the subject of invariant theory is seen to encompass the theory of actionsof linear algebraic groups on affine and projective varieties. A distinct strand of invariant theory, going back to theclassical constructive and combinatorial methods of the nineteenth century, has been developed by Gian-Carlo Rotaand his school. A prominent example of this circle of ideas is given by the theory of standard monomials.

26.2 The nineteenth-century origins

The theory of invariants came into existence about the middle of the nineteenth century somewhat like Minerva: agrown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley’s Jovian head.Weyl (1939b, p.489)

Cayley, whose fundamental work establishing “invariant theory” was “On the Theory of Linear Transformations(1845).” In the opening of his paper, Cayley credits an 1841 paper of George Boole, “investigations were suggestedto me by a very elegant paper on the same subject... byMr Boole.” (Boole’s paper was Exposition of a General Theoryof Linear Transformations, Cambridge Mathematical Journal.)Classically, the term “invariant theory” refers to the study of invariant algebraic forms (equivalently, symmetric ten-sors) for the action of linear transformations. This was a major field of study in the latter part of the nineteenthcentury. Current theories relating to the symmetric group and symmetric functions, commutative algebra, modulispaces and the representations of Lie groups are rooted in this area.In greater detail, given a finite-dimensional vector space V of dimension n we can consider the symmetric algebraS(Sr(V)) of the polynomials of degree r over V, and the action on it of GL(V). It is actually more accurate to considerthe relative invariants of GL(V), or representations of SL(V), if we are going to speak of invariants: that is becausea scalar multiple of the identity will act on a tensor of rank r in S(V) through the r-th power 'weight' of the scalar.The point is then to define the subalgebra of invariants I(Sr(V)) for the action. We are, in classical language, lookingat invariants of n-ary r-ics, where n is the dimension of V. (This is not the same as finding invariants of GL(V) onS(V); this is an uninteresting problem as the only such invariants are constants.) The case that was most studied wasinvariants of binary forms where n = 2.Other work included that of Felix Klein in computing the invariant rings of finite group actions on C2 (the binarypolyhedral groups, classified by the ADE classification); these are the coordinate rings of du Val singularities.Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century,is once again at the forefront of mathematics.Kung & Rota (1984, p.27)

The work of David Hilbert, proving that I(V) was finitely presented in many cases, almost put an end to classicalinvariant theory for several decades, though the classical epoch in the subject continued to the final publications ofAlfred Young, more than 50 years later. Explicit calculations for particular purposes have been known in moderntimes (for example Shioda, with the binary octavics).

26.3 Hilbert’s theorems

Hilbert (1890) proved that if V is a finite-dimensional representation of the complex algebraic group G = SLn(C)then the ring of invariants of G acting on the ring of polynomials R = S(V) is finitely generated. His proof used theReynolds operator ρ from R to RG with the properties

• ρ(1) = 1

• ρ(a + b) = ρ(a) + ρ(b)

• ρ(ab) = a ρ(b) whenever a is an invariant.

80 CHAPTER 26. INVARIANT THEORY

Hilbert constructed the Reynolds operator explicitly using Cayley’s omega process Ω, though now it is more commonto construct ρ indirectly as follows: for compact groups G, the Reynolds operator is given by taking the average overG, and non-compact reductive groups can be reduced to the case of compact groups using Weyl’s unitarian trick.Given the Reynolds operator, Hilbert’s theorem is proved as follows. The ring R is a polynomial ring so is graded bydegrees, and the ideal I is defined to be the ideal generated by the homogeneous invariants of positive degrees. ByHilbert’s basis theorem the ideal I is finitely generated (as an ideal). Hence, I is finitely generated by finitely manyinvariants of G (because if we are given any – possibly infinite – subset S that generates a finitely generated ideal I,then I is already generated by some finite subset of S). Let i1,...,in be a finite set of invariants of G generating I (asan ideal). The key idea is to show that these generate the ring RG of invariants. Suppose that x is some homogeneousinvariant of degree d > 0. Then

x = a1i1 + ... + a i

for some aj in the ring R because x is in the ideal I. We can assume that aj is homogeneous of degree d − deg ij forevery j (otherwise, we replace aj by its homogeneous component of degree d − deg ij; if we do this for every j, theequation x = a1i1 + ... + ani will remain valid). Now, applying the Reynolds operator to x = a1i1 + ... + ani gives

x = ρ(a1)i1 + ... + ρ(an)in

We are now going to show that x lies in the R-algebra generated by i1,...,in.First, let us do this in the case when the elements ρ(ak) all have degree less than d. In this case, they are all in theR-algebra generated by i1,...,in (by our induction assumption). Therefore x is also in this R-algebra (since x = ρ(a1)i1+ ... + ρ(a )i ).In the general case, we cannot be sure that the elements ρ(ak) all have degree less than d. But we can replace eachρ(ak) by its homogeneous component of degree d − deg ij. As a result, these modified ρ(ak) are still G-invariants(because every homogeneous component of a G-invariant is a G-invariant) and have degree less than d (since deg ik> 0). The equation x = ρ(a1)i1 + ... + ρ(a )i still holds for our modified ρ(ak), so we can again conclude that x liesin the R-algebra generated by i1,...,in.Hence, by induction on the degree, all elements of RG are in the R-algebra generated by i1,...,in.

26.4 Geometric invariant theory

The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the constructionof a quotient by the group action that should capture invariant information through its coordinate ring. It is a subtletheory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In aseparate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, hasbeen rehabilitated.One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing markedobjects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology,and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles.

26.5 See also

• Gram’s theorem

• invariant theory of finite groups

• representation theory of finite groups

• Molien series

• invariant (mathematics)

26.6. REFERENCES 81

26.6 References[1] Borel, Armand (2001). Essays in the History of Lie groups and algebraic groups. History of Mathematics, Vol. 21.

American mathematical society and London mathematical society. ISBN 978-0821802885.

• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525 Reprinted as Dieudonné, Jean A.;Carrell, James B. (1971), “Invariant theory, old and new”, Advances in Mathematics (Boston, MA: AcademicPress) 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series 296,Cambridge University Press, doi:10.1017/CBO9780511615436, ISBN 978-0-521-52548-0, MR 2004511

• Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge: Cambridge University Press

• Grosshans, Frank D. (1997), Algebraic homogeneous spaces and invariant theory, New York: Springer, ISBN3-540-63628-5

• Kung, Joseph P. S.; Rota, Gian-Carlo (1984), “The invariant theory of binary forms”, American MathematicalSociety. Bulletin. New Series 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904, MR722856

• Hilbert, David (1890), “Ueber die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831

• Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen 42 (3):313, doi:10.1007/BF01444162

• Neusel, Mara D.; and Smith, Larry (2002), Invariant Theory of Finite Groups, Providence, RI: AmericanMathematical Society, ISBN 0-8218-2916-5 A recent resource for learning about modular invariants of finitegroups.

• Olver, Peter J. (1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 0-521-55821-2 An undergraduate level introduction to the classical theory of invariants of binary forms, includingthe Omega process starting at page 87.

• Popov, V.L. (2001), “Invariants, theory of”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Springer, T. A. (1977), Invariant Theory, New York: Springer, ISBN 0-387-08242-5 An older but still usefulsurvey.

• Sturmfels, Bernd (1993), Algorithms in Invariant Theory, New York: Springer, ISBN 0-387-82445-6 A beau-tiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbnerbases.

• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255

• Weyl, Hermann (1939b), “Invariants”, Duke Mathematical Journal 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030

26.7 External links• H. Kraft, C. Procesi, Classical Invariant Theory, a Primer

Chapter 27

Invariants of tensors

In mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficientsof the characteristic polynomial of the tensor A:[1]

p(λ) := det(A− λE)

where E is the identity tensor and λ ∈ C is the polynomial’s indeterminate (it is important to bear in mind that apolynomial’s indeterminate λ may also be a non-scalar as long as power, scaling and adding make sense for it, e.g.,p(A) is legitimate, and in fact, quite useful).The first invariant of an n×n tensor A ( IA ) is the coefficient for λn−1 (coefficient for λn is always 1), the secondinvariant ( IIA ) is the coefficient for λn−2 , etc., the nth invariant is the free term.The definition of the invariants of tensors and specific notations used throughout the article were introduced into thefield of rheology by Ronald Rivlin and became extremely popular there. In fact even the trace of a tensorA is usuallydenoted as IA in the textbooks on rheology.

27.1 Properties

The first invariant (trace) is always the sum of the diagonal components:

IA = A11 +A22 + · · ·+Ann = tr(A)

The nth invariant is just ± detA , the determinant of A (up to sign).The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function ofthe invariants only is also objective.

27.2 Calculation of the invariants of symmetric 3×3 tensors

Most tensors used in engineering are symmetric 3×3. For this case the invariants can be calculated as:

IA = tr(A) = A11 +A22 +A33 = A1 +A2 +A3

IIA =1

2

((trA)2 − tr(AA)

)= A11A22 +A22A33 +A11A33 −A12A21 −A23A32 −A13A31

(the sum of principal minors)

= A1A2 +A2A3 +A1A3

IIIA = det(A) = A1A2A3

82

27.3. ENGINEERING APPLICATION 83

where A1 , A2 , A3 are the eigenvalues of tensor A.Because of the Cayley–Hamilton theorem the following equation is always true:

A3 − IAA2 + IIAA− IIIAE = 0

where E is the second-order identity tensor.A similar equation holds for tensors of higher order.

27.3 Engineering application

A scalar valued tensor function f that depends merely on the three invariants of a symmetric 3×3 tensorA is objective,i.e., independent from rotations of the coordinate system. Moreover, every objective tensor function depends only onthe tensor’s invariants. Thus, objectivity of a tensor function is fulfilled if, and only if, for some function g : R3 → Rwe have

f(A) = g(IA, IIA, IIIA).

A common application to this is the evaluation of the potential energy as function of the strain tensor, within theframework of linear elasticity. Exhausting the above theorem the free energy of the system reduces to a function of3 scalar parameters rather than 6. Within linear elasticity the free energy has to be quadratic in the tensor’s elements,which eliminates an additional scalar. Thus, for an isotropic material only two independent parameters are neededto describe the elastic properties, known as Lamé coefficients. Consequently, experimental fits and computationalefforts may be eased significantly.

27.4 See also• Symmetric polynomial

• Elementary symmetric polynomial

• Newton’s identities

• Invariant theory

27.5 References[1] SPENCER, A. J. M. Continuum Mechanics. Longman, 1980.

Chapter 28

Kempf–Ness theorem

In algebraic geometry, the Kempf–Ness theorem, introduced by Kempf and Ness (1979), gives a criterion for thestability of a vector in a representation of a complex reductive group. If the complex vector space is given a normthat is invariant under a maximal compact subgroup of the reductive group, then the Kempf–Ness theorem states thata vector is stable if and only if the norm attains a minimum value on the orbit of the vector.

28.1 References• Kempf, George; Ness, Linda (1979), “The length of vectors in representation spaces”, Algebraic geometry(Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math. 732, Berlin, NewYork: Springer-Verlag, pp. 233–243, doi:10.1007/BFb0066647, MR 555701

84

Chapter 29

Kostant polynomial

In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring ofpolynomials over the ring of polynomials invariant under the finite reflection group of a root system.

29.1 Background

If the reflection groupW corresponds to theWeyl group of a compact semisimple groupK with maximal torus T, thenthe Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifold K/T, alsoisomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup. Armand Borelshowed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by theinvariant homogeneous polynomials of positive degree. This ring had already been considered by Claude Chevalleyin establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with AndréWeil, Jean-Louis Koszul and Henri Cartan; the existence of such a basis was used by Chevalley to prove that thering of invariants was itself a polynomial ring. A detailed account of Kostant polynomials was given by Bernstein,Gelfand & Gelfand (1973) and independently Demazure (1973) as a tool to understand the Schubert calculus of theflag manifold. The Kostant polynomials are related to the Schubert polynomials defined combinatorially by Lascoux& Schützenberger (1982) for the classical flag manifold, when G = SL(n,C). Their structure is governed by differenceoperators associated to the corresponding root system.Steinberg (1975) defined an analogous basis when the polynomial ring is replaced by the ring of exponentials of theweight lattice. If K is simply connected, this ring can be identified with the representation ring R(T) and the W-invariant subring with R(K). Steinberg’s basis was again motivated by a problem on the topology of homogeneousspaces; the basis arises in describing the T-equivariant K-theory of K/T.

29.2 Definition

Let Φ be a root system in a finite-dimensional real inner product space V with Weyl group W. Let Φ+ be a set ofpositive roots and Δ the corresponding set of simple roots. If α is a root, then sα denotes the corresponding reflectionoperator. Roots are regarded as linear polynomials on V using the inner product α(v) = (α,v). The choice of Δ givesrise to a Bruhat order on the Weyl group determined by the ways of writing elements minimally as products of simpleroot reflection. The minimal length for an elenet s is denoted ℓ(s) . Pick an element v in V such that α(v) > 0 forevery positive root.If αi is a simple root with reflection operator si

six = x− 2(x, αi)

(αi, αi)αi,

then the corresponding divided difference operator is defined by

85

86 CHAPTER 29. KOSTANT POLYNOMIAL

δif =f − f si

αi.

If ℓ(s) = m and s has reduced expression

s = si1 · · · sim ,

then

δs = δi1 · · · δim

is independent of the reduced expression. Moreover

δsδt = δst

if ℓ(st) = ℓ(s) + ℓ(t) and 0 otherwise.If w0 is the longest element ofW, the element of greatest length or equivalently the element sending Φ+ to −Φ+, then

δw0f =

∑s∈W det s f s∏

α>0 α.

More generally

δsf =det s f s+

∑t<s as,t f t∏

α>0, s−1α<0 α

for some constants as,t.Set

d = |W |−1∏α>0

α.

and

Ps = δs−1w0d.

Then P is a homogeneous polynomial of degree ℓ(s) .These polynomials are the Kostant polynomials.

29.3 Properties

Theorem. The Kostant polynomials form a free basis of the ring of polynomials over the W-invariant polynomials.

In fact the matrix

Nst = δs(Pt)

is unitriangular for any total order such that s ≥ t implies ℓ(s) ≥ ℓ(t) .

29.3. PROPERTIES 87

Hence

detN = 1.

Thus if

f =∑s

asPs

with as invariant underW, then

δt(f) =∑s

δt(Ps)as.

Thus

as =∑t

Ms,tδt(f),

where

M = N−1

another unitriangular matrix with polynomial entries. It can be checked directly that as is invariant underW.In fact δi satisfies the derivation property

δi(fg) = δi(f)g + (f si)δi(g).

Hence

δiδs(f) =∑t

δi(δs(Pt))at) =∑t

(δs(Pt) si)δi(at) +∑t

δiδs(Pt)at.

Since

δiδs = δsis

or 0, it follows that

∑t

δs(Pt) δi(at) si = 0

so that by the invertibility of N

δi(at) = 0

for all i, i.e. at is invariant underW.

88 CHAPTER 29. KOSTANT POLYNOMIAL

29.4 Steinberg basis

As above let Φ be a root system in a real inner product space V, and Φ+ a subset of positive roots. From these datawe obtain the subset Δ = α1, α2, ..., αn of the simple roots, the coroots

α∨i = 2(αi, αi)

−1αi,

and the fundamental weights λ1, λ2, ..., λn as the dual basis of the coroots.For each element s inW, let Δs be the subset of Δ consisting of the simple roots satisfying s−1α < 0, and put

λs = s−1∑αi∈∆s

λi,

where the sum is calculated in the weight lattice P.The set of linear combinations of the exponentials eμ with integer coefficients for μ in P becomes a ring over Zisomorphic to the group algebra of P, or equivalently to the representation ring R(T) of T, where T is a maximal torusin K, the simply connected, connected compact semisimple Lie group with root system Φ. IfW is the Weyl group ofΦ, then the representation ring R(K) of K can be identified with R(T)W .Steinberg’s theorem. The exponentials λs (s in W) form a free basis for the ring of exponentials over the subring ofW-invariant exponentials.Let ρ denote the half sum of the positive roots, and A denote the antisymmetrisation operator

A(ψ) =∑s∈W

(−1)ℓ(s)s · ψ.

The positive roots β with sβ positive can be seen as a set of positive roots for a root system on a subspace of V; theroots are the ones orthogonal to s.λs. The corresponding Weyl group equals the stabilizer of λs inW. It is generatedby the simple reflections sj for which sαj is a positive root.Let M and N be the matrices

Mts = t(λs), Nst = (−1)ℓ(t) · t(ψs),

where ψs is given by the weight s−1ρ - λs. Then the matrix

Bs,s′ = Ω−1(NM)s,s′ =A(ψsλs′)

Ω

is triangular with respect to any total order onW such that s ≥ t implies ℓ(s) ≥ ℓ(t) . Steinberg proved that the entriesof B areW-invariant exponential sums. Moreover its diagonal entries all equal 1, so it has determinant 1. Hence itsinverse C has the same form. Define

φs =∑

Cs,tψt.

If χ is an arbitrary exponential sum, then it follows that

χ =∑s∈W

asλs

with as theW-invariant exponential sum

29.5. REFERENCES 89

as =A(φsχ)

Ω.

Indeed this is the unique solution of the system of equations

tχ =∑s∈W

t(λs) as =∑s

Mt,sas.

29.5 References• Bernstein, I. N.; Gelfand, I. M.; Gelfand, S. I. (1973), “Schubert cells, and the cohomology of the spaces G/P”,Russian Math. Surveys 28: 1–26, doi:10.1070/RM1973v028n03ABEH001557

• Billey, Sara C. (1999), “Kostant polynomials and the cohomology ring for G/B.”, Duke Math. J. 96: 205–224,doi:10.1215/S0012-7094-99-09606-0

• Bourbaki, Nicolas (1981), Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Masson, ISBN 2-225-76076-4

• Cartan, Henri (1950), “Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opèreun groupe de Lie”, Colloque de topologie (espaces fibrés), Bruxelles: 15–27

• Cartan, Henri (1950), “La transgression dans un groupe de Lie et dans un espace fibré principal”, Colloque detopologie (espaces fibrés), Bruxelles: 57–71

• Chevalley, Claude (1955), “Invariants of finite groups generated by reflections”, Amer. J. Math. (The JohnsHopkins University Press) 77 (4): 778–782, doi:10.2307/2372597, JSTOR 2372597

• Demazure, Michel (1973), “Invariants symétriques entiers des groupes de Weyl et torsion”, Invent. Math. 21:287–301, doi:10.1007/BF01418790

• Greub, Werner; Halperin, Stephen; Vanstone, Ray (1976), Connections, curvature, and cohomology. VolumeIII: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, 47-III, Aca-demic Press

• Humphreys, James E. (1994), Introduction to Lie Algebras and Representation Theory (2nd ed.), Springer,ISBN 0-387-90053-5

• Kostant, Bertram (1963), “Lie algebra cohomology and generalized Schubert cells”, Ann. Of Math. (Annalsof Mathematics) 77 (1): 72–144, doi:10.2307/1970202, JSTOR 1970202

• Kostant, Bertram (1963), “Lie group representations on polynomial rings”,Amer. J. Math. (The Johns HopkinsUniversity Press) 85 (3): 327–404, doi:10.2307/2373130, JSTOR 2373130

• Kostant, Bertram; Kumar, Shrawan (1986), “The nil Hecke ring and cohomology of G/P for a Kac–Moodygroup G.”, Proc. Nat. Acad. Sci. U.S.A. 83: 1543–1545, doi:10.1073/pnas.83.6.1543

• Alain, Lascoux; Schützenberger, Marcel-Paul (1982), “Polynômes de Schubert [Schubert polynomials]", C. R.Acad. Sci. Paris Sér. I Math. 294: 447–450

• McLeod, John (1979), The Kunneth formula in equivariant K-theory, Lecture Notes in Math. 741, Springer,pp. 316–333

• Steinberg, Robert (1975), “On a theorem of Pittie”, Topology 14: 173–177, doi:10.1016/0040-9383(75)90025-7

Chapter 30

Littlewood–Richardson rule

In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise whendecomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients arenatural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur inmany other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of irreduciblerepresentations of general linear groups (or related groups like the special linear and special unitary groups), or in thedecomposition of certain induced representations in the representation theory of the symmetric group, or in the areaof algebraic combinatorics dealing with Young tableaux and symmetric polynomials.Littlewood–Richardson coefficients depend on three partitions, say λ, µ, ν , of which λ and µ describe the Schurfunctions being multiplied, and ν gives the Schur function of which this is the coefficient in the linear combination;in other words they are the coefficients cνλ,µ such that

sλsµ =∑ν

cνλ,µsν .

The Littlewood–Richardson rule states that cνλ,µ is equal to the number of Littlewood–Richardson tableaux of skewshape ν/λ and of weight µ .

30.1 History

Unfortunately the Littlewood–Richardson rule is much harder to prove than was at first suspected. The author wasonce told that the Littlewood–Richardson rule helped to get men on the moon but was not proved until after they gotthere.Gordon James (1987)

The Littlewood–Richardson rule was first stated by D. E. Littlewood and A. R. Richardson (1934, theorem III p.119) but though they claimed it as a theorem they only proved it in some fairly simple special cases. Robinson (1938)claimed to complete their proof, but his argument had gaps, though it was so obscurely written that these gaps werenot noticed for some time, and his argument is reproduced in the book (Littlewood 1950). Some of the gaps werelater filled by Macdonald (1995). The first rigorous proofs of the rule were given four decades after it was found, bySchützenberger (1977) and Thomas (1974), after the necessary combinatorial theory was developed by C. Schensted(1961), Schützenberger (1963), and Knuth (1970) in their work on the Robinson–Schensted correspondence. Thereare now several short proofs of the rule, such as (Gasharov 1998), and (Stembridge 2002) using Bender-Knuth in-volutions. Littelmann (1994) used the Littelmann path model to generalize the Littlewood–Richardson rule to othersemisimple Lie groups.The Littlewood–Richardson rule is notorious for the number of errors that appeared prior to its complete, publishedproof. Several published attempts to prove it are incomplete, and it is particularly difficult to avoid errors when doinghand calculations with it: even the original example in D. E. Littlewood and A. R. Richardson (1934) contains anerror.

90

30.1. HISTORY 91

30.1.1 Littlewood–Richardson tableaux

11 121

32A Littlewood–Richardson tableau

A Littlewood–Richardson tableau is a skew semistandard tableau with the additional property that the sequenceobtained by concatenating its reversed rows is a lattice word (or lattice permutation), which means that in every initialpart of the sequence any number i occurs at least as often as the number i + 1 . Another equivalent (though notquite obviously so) characterization is that the tableau itself, and any tableau obtained from it by removing somenumber of its leftmost columns, has a weakly decreasing weight. Many other combinatorial notions have been foundthat turn out to be in bijection with Littlewood–Richardson tableaux, and can therefore also be used to define theLittlewood–Richardson coefficients.

30.1.2 Example

Consider the case that λ = (2, 1) , µ = (3, 2, 1) and ν = (4, 3, 2) . Then the fact that cνλ,µ = 2 can be deducedfrom the fact that the two tableaux shown at the right are the only two Littlewood–Richardson tableaux of shape ν/λand weight µ . Indeed, since the last box on the first nonempty line of the skew diagram can only contain an entry 1,the entire first line must be filled with entries 1 (this is true for any Littlewood–Richardson tableau); in the last boxof the second row we can only place a 2 by column strictness and the fact that our lattice word cannot contain anylarger entry before it contains a 2. For the first box of the second row we can now either use a 1 or a 2. Once thatentry is chosen, the third row must contain the remaining entries to make the weight (3,2,1), in a weakly increasingorder, so we have no choice left any more; in both case it turns out that we do find a Littlewood–Richardson tableau.

30.1.3 A more geometrical description

The condition that the sequence of entries read from the tableau in a somewhat peculiar order form a lattice wordcan be replaced by a more local and geometrical condition. Since in a semistandard tableau equal entries never occurin the same column, one can number the copies of any value from right to left, which is their order of occurrencein the sequence that should be a lattice word. Call the number so associated to each entry its index, and write anentry i with index j as i[j]. Now if some Littlewood–Richardson tableau contains an entry i > 1 with index j, thenthat entry i[j] should occur in a row strictly below that of (i − 1)[j] (which certainly also occurs, since the entry i− 1 occurs as least as often as the entry i does). In fact the entry i[j] should also occur in a column no further tothe right than that same entry (i − 1)[j] (which at first sight appears to be a stricter condition). If the weight of theLittlewood–Richardson tableau is fixed beforehand, then one can form a fixed collection of indexed entries, and if

92 CHAPTER 30. LITTLEWOOD–RICHARDSON RULE

11 122

31Another Littlewood–Richardson tableau

these are placed in a way respecting those geometric restrictions, in addition to those of semistandard tableaux andthe condition that indexed copies of the same entries should respect right-to-left ordering of the indexes, then theresulting tableaux are guaranteed to be Littlewood–Richardson tableaux.

30.2 An algorithmic form of the rule

The Littlewood–Richardson as stated above gives a combinatorial expression for individual Littlewood–Richardsoncoefficients, but gives no indication of a practical method to enumerate the Littlewood–Richardson tableaux in orderto find the values of these coefficients. Indeed for given λ, µ, ν there is no simple criterion to determine whether anyLittlewood–Richardson tableaux of shape ν/λ and of weight µ exist at all (although there are a number of necessaryconditions, the simplest of which is |λ| + |µ| = |ν| ); therefore it seems inevitable that in some cases one has to gothrough an elaborate search, only to find that no solutions exist.Nevertheless, the rule leads to a quite efficient procedure to determine the full decomposition of a product of Schurfunctions, in other words to determine all coefficients cνλ,µ for fixed λ and μ, but varying ν. This fixes the weight of theLittlewood–Richardson tableaux to be constructed and the “inner part” λ of their shape, but leaves the “outer part” νfree. Since the weight is known, the set of indexed entries in the geometric description is fixed. Now for successiveindexed entries, all possible positions allowed by the geometric restrictions can be tried in a backtracking search. Theentries can be tried in increasing order, while among equal entries they can be tried by decreasing index. The latterpoint is the key to efficiency of the search procedure: the entry i[j] is then restricted to be in a column to the rightof i[j + 1] , but no further to the right than i − 1[j] (if such entries are present). This strongly restricts the set ofpossible positions, but always leaves at least one valid position for i[j] ; thus every placement of an entry will giverise to at least one complete Littlewood–Richardson tableau, and the search tree contains no dead ends.A similar method can be used to find all coefficients cνλ,µ for fixed λ and ν, but varying μ.

30.3 Littlewood–Richardson coefficients

The Littlewood–Richardson coefficients cνλμ appear in the following ways:

30.4. GENERALIZATIONS AND SPECIAL CASES 93

• They are the structure constants for the product in the ring of symmetric functions with respect to the basis ofSchur functions

sλsµ =∑cνλµsν

or equivalently cνλμ is the inner product of sν and sλsμ.

• They express skew Schur functions in terms of Schur functions

sν/λ =∑µ

cνλµsµ.

• The cνλμ appear as intersection numbers on a Grassmannian:

σλσµ =∑cνλµσν

where σμ is the class of the Schubert variety of a Grassmannian corresponding to μ.

• cνλμ is the number of times the irreducible representation Vλ ⊗ Vμ of the product of symmetric groups S|λ| ×S|μ| appears in the restriction of the representation Vν of S|ν| to S|λ| × S|μ|. By Frobenius reciprocity this isalso the number of times that Vν occurs in the representation of S|ν| induced from Vλ ⊗ Vμ.

• The cνλμ appear in the decomposition of the tensor product (Fulton 1997) of two Schur modules (irreducible repre-sentations of special linear groups)

Eλ ⊗ Eµ =⊕ν

(Eν)⊕cνλµ .

• cνλμ is the number of standard Young tableaux of shape ν/μ that are jeu de taquin equivalent to some fixedstandard Young tableau of shape λ.

• cνλμ is the number of Littlewood–Richardson tableaux of shape ν/λ and of weight μ.

• cνλμ is the number of pictures between μ and ν/λ.

30.4 Generalizations and special cases

Zelevinsky (1981) extended the Littlewood–Richardson rule to skew Schur functions as follows:

sλsµ/ν =∑

λ+ω(T≥j)∈P

sλ+ω(T )

where the sum is over all tableaux T on μ/ν such that for all j, the sequence of integers λ+ω(T≥j) is non-increasing,and ω is the weight.Pieri’s formula, which is the special case of the Littlewood–Richardson rule in the case when one of the partitionshas only one part, states that

• SµSn =∑λ

Sλ

94 CHAPTER 30. LITTLEWOOD–RICHARDSON RULE

where Sn is the Schur function of a partition with one row and the sum is over all partitions λ obtained from μ byadding n elements to its Ferrers diagram, no two in the same column.If both partitions are rectangular in shape, the sum is also multiplicity free (Okada 1998). Fix a, b, p, and qpositive integers with p ≥ q. Denote by (ap) the partition with p parts of length a. The partitions indexing nontrivialcomponents of s(ap)s(bq) are those partitions λ with length ≤ p+ q such that

• λq+1 = λq+2 = · · · = λp = a,

• λq ≥ max(a, b)

• λi + λp+q−i+1 = a+ b, i = 1, . . . , q.

For example,

.

30.5 Examples

The examples of Littlewood-Richardson coefficients below are given in terms of products of Schur polynomials Sπ,indexed by partitions π, using the formula

SλSµ =∑

cνλµSν .

All coefficients with ν at most 4 are given by:

• S0Sπ = Sπ for any π. where S0=1 is the Schur polynomial of the empty partition

• S1S1 = S2 + S11• S2S1 = S3 + S21• S11S1 = S111 + S21• S3S1 = S4 + S31• S21S1 = S31 + S22 + S211• S2S2 = S4 + S31 + S22• S2S11 = S31 + S211

30.6. REFERENCES 95

• S111S1 = S1111 + S211• S11S11 = S1111 + S211 + S22

Most of the coefficients for small partitions are 0 or 1, which happens in particular whenever one of the factors isof the form Sn or S₁₁...₁, because of Pieri’s formula and its transposed counterpart. The simplest example with acoefficient larger than 1 happens when neither of the factors has this form:

• S21S21 = S42 + S411 + S33 + 2S321 + S3111 + S222 + S2211.

For larger partitions the coefficients become more complicated. For example,

• S321S321 = S642 +S6411 +S633 +2S6321 +S63111 +S6222 +S62211 +S552 +S5511 +2S543 +4S5421 +2S54111 +3S5331+3S5322 +4S53211 +S531111 +2S52221 +S522111 +S444 +3S4431 +2S4422 +3S44211 +S441111 +3S4332 +3S43311+4S43221 +2S432111 +S42222 +S422211 +S3333 +2S33321 +S333111 +S33222 +S332211 with 34 terms and totalmultiplicity 62, and the largest coefficient is 4

• S4321S4321 is a sum of 206 terms with total multiplicity is 930, and the largest coefficient is 18.

• S54321S54321 is a sum of 1433 terms with total multiplicity 26704, and the largest coefficient (that of S86543211)is 176.

• S654321S654321 is a sum of 10873 terms with total multiplicity is 1458444 (so the average value of the coeffi-cients is more than 100, and they can be as large as 2064).

The original example given by Littlewood & Richardson (1934, p. 122-124) was (after correcting for 3 tableaux theyfound but forgot to include in the final sum)

• S431S221 = S652 + S6511 + S643 + 2S6421 + S64111 + S6331 + S6322 + S63211 + S553 + 2S5521 + S55111 + 2S5431+ 2S5422 + 3S54211 + S541111 + S5332 + S53311 + 2S53221 + S532111 + S4432 + S44311 + 2S44221 + S442111 +S43321 + S43222 + S432211

with 26 terms coming from the following 34 tableaux:....11 ....11 ....11 ....11 ....11 ....11 ....11 ....11 ....11 ...22 ...22 ...2 ...2 ...2 ...2 ... ... ... .3 . .23 .2 .3 . .22 .2 .2 3 3 2 23 23 2 3 3 ....1 ....1 ....1 ....1 ....1 ....1 ....1 ....1 ....1 ...12 ...12 ...12 ...12 ...1 ...1 ...1 ...2 ...1 .23 .2 .3 . .23 .22 .2 .1 .23 2 2 2 3 23 23 2 3 3 ....1 ....1 ....1 ....1 ....1 ....1 ....1 ....1 ...2 ...2 ...2 ... ... ... ... ... .1 .3 . .12 .12 .1 .2 .2 2 1 1 23 222 13 1 3 2 2 3 3 2 2 3 3 .... .... .... .... .... .... .... .... ...1 ...1 ...1 ...1 ...1 ... ... ... .12 .12 .1 .2 .2 .11 .1 .1 23 2 22 131 22 12 12 3 3 2 2 3 23 2 3 3Calculating skew Schur functions is similar. For example, the 15 Littlewood–Richardson tableaux for ν=5432 andλ=331 are...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2...2 ...2 ...2 ...2 ...2 .11 .11 .11 .12 .11 .12 .13 .13 .23 .13 .13 .12 .12 .23 .23 12 13 22 12 23 13 12 24 14 14 22 2333 13 34so S₅₄₃₂/₃₃₁ = Σcνλμ Sμ = S52 + S511 + S4111 + S2221 + 2S43 + 2S3211 + 2S322 + 2S331 + 3S421 (Fulton 1997, p. 64).

30.6 References• Fulton, William (1997), Young tableaux, London Mathematical Society Student Texts 35, Cambridge Univer-sity Press, p. 121, ISBN 978-0-521-56144-0, MR 1464693

• Gasharov, Vesselin (1998), “A short proof of the Littlewood-Richardson rule”, European Journal of Combi-natorics 19 (4): 451–453, doi:10.1006/eujc.1998.0212, ISSN 0195-6698, MR 1630540

• James, Gordon (1987), “The representation theory of the symmetric groups”, The Arcata Conference on Repre-sentations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math. 47, Providence, R.I.: AmericanMathematical Society, pp. 111–126, MR 933355

96 CHAPTER 30. LITTLEWOOD–RICHARDSON RULE

• Knuth, Donald E. (1970), “Permutations, matrices, and generalized Young tableaux”, Pacific Journal of Math-ematics 34: 709–727, doi:10.2140/pjm.1970.34.709, ISSN 0030-8730, MR 0272654

• Littelmann, Peter (1994), “A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras” (PDF),Invent. Math. 116: 329–346, doi:10.1007/BF01231564

• Littlewood, Dudley E. (1950), The theory of group characters and matrix representations of groups, AMSChelsea Publishing, Providence, RI, ISBN 978-0-8218-4067-2, MR 00002127

• Littlewood, D. E.; Richardson, A. R. (1934), “Group Characters and Algebra”, Philosophical Transactions ofthe Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character (The RoyalSociety) 233 (721–730): 99–141, doi:10.1098/rsta.1934.0015, ISSN 0264-3952

• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nded.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144

• Okada, Soichi (1998), “Applications of minor summation formulas to rectangular-shaped representations ofclassical groups”, Journal of Algebra 205 (2): 337–367, doi:10.1006/jabr.1997.7408, ISSN 0021-8693, MR1632816

• Robinson, G. de B. (1938), “On the Representations of the Symmetric Group”,American Journal ofMathemat-ics (The Johns Hopkins University Press) 60 (3): 745–760, doi:10.2307/2371609, ISSN 0002-9327, JSTOR2371609 Zbl0019.25102

• Schensted, C. (1961), “Longest increasing and decreasing subsequences”, Canadian Journal of Mathematics13: 179–191, doi:10.4153/CJM-1961-015-3, ISSN 0008-414X, MR 0121305

• Schützenberger, M. P. (1963), “Quelques remarques sur une construction de Schensted”, Mathematica Scan-dinavica 12: 117–128, ISSN 0025-5521, MR 0190017

• Schützenberger, Marcel-Paul (1977), “La correspondance de Robinson”, Combinatoire et représentation dugroupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) (PDF), Lecturenotes in mathematics 579, Berlin, New York: Springer-Verlag, pp. 59–113, doi:10.1007/BFb0090012, ISBN978-3-540-08143-2, MR 0498826

• Stembridge, John R. (2002), “A concise proof of the Littlewood-Richardson rule” (PDF), Electronic Journalof Combinatorics 9 (1): Note 5, 4 pp. (electronic), ISSN 1077-8926, MR 1912814

• Thomas, Glânffrwd P. (1974), Baxter algebras and Schur functions, Ph.D. Thesis, Swansea: University Collegeof Swansea

• van Leeuwen, Marc A. A. (2001), “The Littlewood-Richardson rule, and related combinatorics”, Interactionof combinatorics and representation theory (PDF), MSJ Mem. 11, Tokyo: Math. Soc. Japan, pp. 95–145, MR1862150

• Zelevinsky, A. V. (1981), “A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence”, Journal of Algebra 69 (1): 82–94, doi:10.1016/0021-8693(81)90128-9, ISSN 0021-8693, MR 613858

30.7 External links• An online program, decomposing products of Schur functions using the Littlewood–Richardson rule

Chapter 31

Modular invariant theory

In mathematics, amodular invariant of a group is an invariant of a finite group acting on a vector space of positivecharacteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914by Dickson (2004).

31.1 Dickson invariant

When G is the finite general linear group GLn(Fq) over the finite field Fq of order a prime power q acting on the ringFq[X1, ...,Xn] in the natural way, Dickson (1911) found a complete set of invariants as follows. Write [e1, ...,en] forthe determinant of the matrix whose entries are Xqeji, where e1, ...,en are non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is

∣∣∣∣∣∣∣x1 xq1 xq

2

1

x2 xq2 xq2

2

x3 xq3 xq2

3

∣∣∣∣∣∣∣Then under the action of an element g of GLn(Fq) these determinants are all multiplied by det(g), so they are allinvariants of SLn(Fp) and the ratios [e1, ...,en]/[0, 1, ...,n − 1] are invariants of GLn(Fq), calledDickson invariants.Dickson proved that the full ring of invariants Fq[X1, ...,Xn]GLn(Fq) is a polynomial algebra over the n Dicksoninvariants [0, 1, ...,i − 1, i + 1, ..., n]/[0,1,...,n−1] for i = 0, 1, ..., n − 1. Steinberg (1987) gave a shorter proof ofDickson’s theorem.The matrices [e1, ...,en] are divisible by all non-zero linear forms in the variables Xi with coefficients in the finite fieldFq. In particular the Moore determinant [0, 1, ..., n − 1] is a product of such linear forms, taken over 1 + q + q2 +... + qn – 1 representatives of (n – 1)-dimensional projective space over the field. This factorization is similar to thefactorization of the Vandermonde determinant into linear factors.

31.2 See also• Miss Sanderson’s theorem

31.3 References• Dickson, Leonard Eugene (1911), “A Fundamental System of Invariants of the General Modular Linear Groupwith a Solution of the Form Problem”, Transactions of the American Mathematical Society (Providence, R.I.:American Mathematical Society) 12 (1): 75–98, ISSN 0002-9947, JSTOR 1988736

• Dickson, Leonard Eugene (2004) [1914], On invariants and the theory of numbers, Dover Phoenix editions,New York: Dover Publications, ISBN 978-0-486-43828-3, MR 0201389

97

98 CHAPTER 31. MODULAR INVARIANT THEORY

• Rutherford, Daniel Edwin (2007) [1932], Modular invariants, Cambridge Tracts in Mathematics and Mathe-matical Physics, No. 27, Ramsay Press, ISBN 978-1-4067-3850-6, MR 0186665

• Sanderson, Mildred (1913), “Formal Modular Invariants with Application to Binary Modular Covariants”,Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 14(4): 489–500, ISSN 0002-9947, JSTOR 1988702

• Steinberg, Robert (1987), “On Dickson’s theorem on invariants”, Journal of the Faculty of Science. Universityof Tokyo. Section IA. Mathematics 34 (3): 699–707, ISSN 0040-8980, MR 927606

Chapter 32

Moduli space

In algebraic geometry, amoduli space is a geometric space (usually a scheme or an algebraic stack) whose points rep-resent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequentlyarise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smoothalgebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize suchobjects by introducing coordinates on the resulting space. In this context, the term “modulus” is used synonymouslywith “parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects.

32.1 Motivation

Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli spacecorrespond to solutions of geometric problems. Here different solutions are identified if they are isomorphic (thatis, geometrically the same). Moduli spaces can be thought of as giving a universal space of parameters for theproblem. For example, consider the problem of finding all circles in the Euclidean plane up to congruence. Anycircle can be described uniquely by giving three points, but many different sets of three points give the same circle:the correspondence is many-to-one. However, circles are uniquely parameterized by giving their center and radius:this is two real parameters and one positive real parameter. Since we are only interested in circles “up to congruence”,we identify circles having different centers but the same radius, and so the radius alone suffices to parameterize theset of interest. The moduli space is therefore the positive real numbers.Moduli spaces often carry natural geometric and topological structures as well. In the example of circles, for instance,the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric fordetermining when two circles are “close”. The geometric structure of moduli spaces locally tells us when two solutionsof a geometric classification problem are “close”, but generally moduli spaces also have a complicated global structureas well.For example, consider how to describe the collection of lines in R2 which intersect the origin. We want to assign aquantity, a modulus, to each line L of this family that can uniquely identify it, for example a positive angle θ(L) with0 ≤ θ < π radians, which will yield all lines in R2 which intersect the origin. The set of lines L just constructed isknown as P1(R) and is called the real projective line.We can also describe the collection of lines in R2 that intersect the origin by means of a topological construction.That is, consider S1 ⊂ R2 and notice that to every point s ∈ S1 that we can identify a line L(s) in the collection if theline intersects the origin and s. Yet, this map is two-to-one, so we want to identify s ~ −s to yield P1(R) ≅ S1/~ wherethe topology on this space is the quotient topology induced by the quotient map S1 → P1(R).Thus, when we consider P1(R) as a moduli space of lines that intersect the origin inR2, we capture the ways in whichthe members of the family (lines in the case) can modulate by continuously varying 0 ≤ θ < π.

32.2 Basic examples

99

100 CHAPTER 32. MODULI SPACE

θs

s

L L

Constructing P1(R) by varying 0 ≤ θ < π or as a quotient space of S1.

32.2.1 Projective space and Grassmannians

The real projective space Pn is a moduli space which parametrizes the space of lines in Rn+1 which pass through theorigin. Similarly, complex projective space is the space of all complex lines in Cn+1 passing through the origin.More generally, the GrassmannianG(k, V) of a vector space V over a field F is the moduli space of all k-dimensionallinear subspaces of V.

32.2.2 Chow variety

The Chow variety Chow(d,P3) is a projective algebraic variety which parametrizes degree d curves in P3. It isconstructed as follows. Let C be a curve of degree d in P3, then consider all the lines in P3 that intersect the curve C.This is a degree d divisor D_C in G(2, 4) the Grassmannian of lines in P3. When C varies, by associating C to D_C,we obtain a parameter space of degree d curves as a subset of the space of degree d divisors of the Grassmannian:Chow(d,P3).

32.2.3 Hilbert scheme

The Hilbert scheme Hilb(X) is a moduli scheme. Every closed point of Hilb(X) corresponds to a closed subschemeof a fixed scheme X, and every closed subscheme is represented by such a point.

32.3 Definitions

There are several related notions of things we could call moduli spaces. Each of these definitions formalizes a differentnotion of what it means for the points of a space M to represent geometric objects.

32.3.1 Fine moduli spaces

This is the standard concept. Heuristically, if we have a space M for which each point m∈ M corresponds to analgebro-geometric object Um, then we can assemble these objects into a topological family U overM. (For example,the GrassmannianG(k, V) carries a rank k bundle whose fiber at any point [L] ∈G(k, V) is simply the linear subspaceL ⊂ V.) M is called a base space of the family U. We say that such a family is universal if any family of algebro-geometric objects T over any base space B is the pullback of U along a unique map B→M. A fine moduli space is aspace M which is the base of a universal family.

32.4. FURTHER EXAMPLES 101

More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of allsuitable families of objects with base B. A spaceM is a fine moduli space for the functor F ifM corepresents F, i.e.,there is a natural isomorphism τ : F → Hom(−, M), where Hom(−, M) is the functor of points. This implies that Mcarries a universal family; this family is the family on M corresponding to the identity map 1M ∈ Hom(M, M).

32.3.2 Coarse moduli spaces

Fine moduli spaces are desirable, but they do not always exist and are frequently difficult to construct, so mathe-maticians sometimes use a weaker notion, the idea of a coarse moduli space. A space M is a coarse moduli spacefor the functor F if there exists a natural transformation τ : F → Hom(−, M) and τ is universal among such naturaltransformations. More concretely,M is a coarse moduli space for F if any family T over a base B gives rise to a mapφT : B→ M and any two objects V andW (regarded as families over a point) correspond to the same point of M ifand only if V and W are isomorphic. Thus, M is a space which has a point for every object that could appear in afamily, and whose geometry reflects the ways objects can vary in families. Note, however, that a coarse moduli spacedoes not necessarily carry any family of appropriate objects, let alone a universal one.In other words, a fine moduli space includes both a base spaceM and universal family U →M, while a coarse modulispace only has the base space M.

32.3.3 Moduli stacks

It is frequently the case that interesting geometric objects come equipped with lots of natural automorphisms. Thisin particular makes the existence of a fine moduli space impossible (intuitively, the idea is that if L is some geometricobject, the trivial family L × [0,1] can be made into a twisted family on the circle S1 by identifying L × 0 with L× 1 via a nontrivial automorphism. Now if a fine moduli space X existed, the map S1 → X should not be constant,but would have to be constant on any proper open set by triviality), one can still sometimes obtain a coarse modulispace. However, this approach is not ideal, as such spaces are not guaranteed to exist, are frequently singular whenthey do exist, and miss details about some non-trivial families of objects they classify.A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely,on any base B one can consider the category of families on B with only isomorphisms between families taken asmorphisms. One then considers the fibred category which assigns to any space B the groupoid of families over B.The use of these categories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61).In general they cannot be represented by schemes or even algebraic spaces, but in many cases they have a naturalstructure of an algebraic stack.Algebraic stacks and their use to analyse moduli problems appeared in Deligne-Mumford (1969) as a tool to provethe irreducibility of the (coarse) moduli space of curves of a given genus. The language of algebraic stacks essentiallyprovides a systematic way to view the fibred category that constitutes the moduli problem as a “space”, and the modulistack of many moduli problems is better-behaved (such as smooth) than the corresponding coarse moduli space.

32.4 Further examples

32.4.1 Moduli of curves

For more details on this topic, see Moduli of algebraic curves.

The moduli stack Mg classifies families of smooth projective curves of genus g, together with their isomorphisms.When g > 1, this stack may be compactified by adding new “boundary” points which correspond to stable nodalcurves (together with their isomorphisms). A curve is stable if it has only a finite group of automorphisms. Theresulting stack is denoted Mg . Both moduli stacks carry universal families of curves. One can also define coarsemoduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actuallystudied before the notion of moduli stack was invented. In fact, the idea of a moduli stack was invented by Deligneand Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has becomeapparent that the stack of curves is actually the more fundamental object.Both stacks above have dimension 3g−3; hence a stable nodal curve can be completely specified by choosing the

102 CHAPTER 32. MODULI SPACE

values of 3g−3 parameters, when g > 1. In lower genus, one must account for the presence of smooth families ofautomorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere,and its group of isomorphisms is PGL(2). Hence, the dimension ofM0 is

dim(space of genus zero curves) - dim(group of automorphisms) = 0 − dim(PGL(2)) = −3.

Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional groupof automorphisms. Hence, the stack M1 has dimension 0. The coarse moduli spaces have dimension 3g−3 as thestacks when g > 1 because the curves with genus g > 1 have only a finite group as its automorphism i.e. dim(groupof automorphisms) = 0. Eventually, in genus zero the coarse moduli space has dimension zero, and in genus one, ithas dimension one.One can also enrich the problem by considering the moduli stack of genus g nodal curves with nmarked points. Suchmarked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite.The resulting moduli stacks of smooth (or stable) genus g curves with n-marked points are denotedMg,n (orMg,n

), and have dimension 3g−3+n.A case of particular interest is the moduli stack M1,1 of genus 1 curves with one marked point. This is the stackof elliptic curves, and is the natural home of the much studied modular forms, which are meromorphic sections ofbundles on this stack.

32.4.2 Moduli of varieties

In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties.This is the problem underlying Siegel modular form theory. See also Shimura variety.

32.4.3 Moduli of vector bundles

Another important moduli problem is to understand the geometry of (various substacks of) the moduli stack Vectn(X)of rank n vector bundles on a fixed algebraic variety X. This stack has been most studied when X is one-dimensional,and especially when n equals one. In this case, the coarse moduli space is the Picard scheme, which like the modulispace of curves, was studied before stacks were invented. Finally, when the bundles have rank 1 and degree zero, thestudy of the coarse moduli space is the study of the Jacobian variety.In applications to physics, the number of moduli of vector bundles and the closely related problem of the number ofmoduli of principal G-bundles has been found to be significant in gauge theory.

32.5 Methods for constructing moduli spaces

The modern formulation of moduli problems and definition of moduli spaces in terms of the moduli functors (ormore generally the categories fibred in groupoids), and spaces (almost) representing them, dates back to Grothendieck(1960/61), in which he described the general framework, approaches and main problems using Teichmüller spacesin complex analytical geometry as an example. The talks in particular describe the general method of constructingmoduli spaces by first rigidifying the moduli problem under consideration.More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to havea fine moduli space. However, it is often possible to consider a modified moduli problem of classifying the originalobjects together with additional data, chosen in such a way that the identity is the only automorphism respecting alsothe additional data. With a suitable choice of the rigidifying data, the modified moduli problem will have a (fine)moduli space T, often described as a subscheme of a suitable Hilbert scheme or Quot scheme. The rigidifying data ismoreover chosen so that it corresponds to a principal bundle with an algebraic structure group G. Thus one can moveback from the rigidified problem to the original by taking quotient by the action ofG, and the problem of constructingthe moduli space becomes that of finding a scheme (or more general space) that is (in a suitably strong sense) thequotient T/G of T by the action of G. The last problem in general does not admit a solution; however, it is addressedby the groundbreaking geometric invariant theory (GIT), developed by David Mumford in 1965, which shows thatunder suitable conditions the quotient indeed exists.

32.6. IN PHYSICS 103

To see how this might work, consider the problem of parametrizing smooth curves of genus g > 2. A smooth curvetogether with a complete linear system of degree d > 2g is equivalent to a closed one dimensional subscheme ofthe projective space Pd−g. Consequently, the moduli space of smooth curves and linear systems (satisfying certaincriteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space. This locus H inthe Hilbert scheme has an action of PGL(n) which mixes the elements of the linear system; consequently, the modulispace of smooth curves is then recovered as the quotient of H by the projective general linear group.Another general approach is primarily associated with Michael Artin. Here the idea is to start with any object of thekind to be classified and study its deformation theory. This means first constructing infinitesimal deformations, thenappealing to prorepresentability theorems to put these together into an object over a formal base. Next an appealto Grothendieck’s formal existence theorem provides an object of the desired kind over a base which is a completelocal ring. This object can be approximated via Artin’s approximation theorem by an object defined over a finitelygenerated ring. The spectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desiredmoduli space. By gluing together enough of these charts, we can cover the space, but the map from our union ofspectra to the moduli space will in general be many to one. We therefore define an equivalence relation on the former;essentially, two points are equivalent if the objects over each are isomorphic. This gives a scheme and an equivalencerelation, which is enough to define an algebraic space (actually an algebraic stack if we are being careful) if not alwaysa scheme.

32.6 In physics

For more details on this topic, see moduli (physics).

The term moduli space is sometimes used in physics to refer specifically to the moduli space of vacuum expectationvalues of a set of scalar fields, or to the moduli space of possible string backgrounds.Moduli spaces also appear in physics in cohomological field theory, where one can use Feynman path integrals tocompute the intersection numbers of various algebraic moduli spaces.

32.7 References• Grothendieck, Alexander (1960–1961). “Techniques de construction en géométrie analytique. I. Descriptionaxiomatique de l'espace de Teichmüller et de ses variantes.” (PDF). Séminaire Henri Cartan 13 no. 1, ExposésNo. 7 and 8 (Paris: Secrétariat Mathématique).

• Mumford, David,Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge,Band 34 Springer-Verlag, Berlin-New York 1965 vi+145 pp MR 0214602

• Mumford, David; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathe-matik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin,1994. xiv+292 pp. MR 1304906 ISBN 3-540-56963-4

• Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMALectures inMathematicsand Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 978-3-03719-029-6, MR2284826

• Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathe-matics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055, ISBN978-3-03719-055-5, MR2524085

• Papadopoulos, Athanase, ed. (2012), Handbook of Teichmüller theory. Vol. III, IRMA Lectures in Mathe-matics and Theoretical Physics, 17, European Mathematical Society (EMS), Zürich, doi:10.4171/103, ISBN978-3-03719-103-3.

• Deligne, Pierre; Mumford, David (1969). “The irreducibility of the space of curves of given genus” (PDF).Publications Mathématiques de l'IHÉS (Paris) 36: 75–109. doi:10.1007/bf02684599.

104 CHAPTER 32. MODULI SPACE

• Harris, Joe; Morrison, Ian (1998). Moduli of Curves. Graduate Texts inMathematics 187. NewYork: SpringerVerlag. doi:10.1007/b98867. ISBN 0-387-98429-1. MR 1631825.

• Katz, Nicholas M; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies108. Princeton University Press. ISBN 0-691-08352-5. MR 772569.

• Faltings, Gerd; Chai, Ching-Li (1990). Degeneration of Abelian Varieties. Ergebnisse der Mathematik undihrer Grenzgebiete 22. With an appendix by David Mumford. Berlin: Springer-Verlag. doi:10.1007/978-3-662-02632-8. ISBN 3-540-52015-5. MR 1083353.

• Viehweg, Eckart (1995). Quasi-Projective Moduli for Polarized Manifolds (PDF). Springer Verlag. ISBN 3-540-59255-5.

• Simpson, Carlos (1994). “Moduli of representations of the fundamental group of a smooth projective varietyI” (PDF). Publications Mathématiques de l'IHÉS (Paris) 79: 47–129. doi:10.1007/bf02698887.

32.8 External links• J. Lurie, Moduli Problems for Ring Spectra

Chapter 33

Molien series

In mathematics, a Molien series is a generating function attached to a linear representation ρ of a group G on afinite-dimensional vector space V. It counts the homogeneous polynomials of a given total degree d that are invariantsfor G. It is named for Theodor Molien.

33.1 Formulation

More formally, there is a vector space of such polynomials, for each given value of d = 0, 1, 2, ..., and we write nd forits vector space dimension, or in other words the number of linearly independent homogeneous invariants of a givendegree. In more algebraic terms, take the d-th symmetric power of V, and the representation of G on it arising fromρ. The invariants form the subspace consisting of all vectors fixed by all elements of G, and nd is its dimension.The Molien series is then by definition the formal power series

M(t) =∑d

ndtd.

This can be looked at another way, by considering the representation of G on the symmetric algebra of V, and thenthe whole subalgebra R of G-invariants. Then nd is the dimension of the homogeneous part of R of dimension d,when we look at it as graded ring. In this way a Molien series is also a kind of Hilbert function. Without furtherhypotheses not a great deal can be said, but assuming some conditions of finiteness it is then possible to show that theMolien series is a rational function. The case of finite groups is most often studied.

33.2 Formula

Molien showed that

M(t) =1

|G|∑g∈G

1

det(I − tg)

This means that the coefficient of td in this series is the dimension nd defined above. It assumes that the characteristicof the field does not divide |G| (but even without this assumption, Molien’s formula in the form |G| · M(t) =∑g∈G

1det(I−tg) is valid, although it does not help with computing M(t)).

33.3 Example

Consider S3 acting on R3 by permuting the coordinates. Note that det(I − tg) is constant on conjugacy classes, so itis enough to take one from each of the three classes in S3 ; so det(I− te) = (1− t)3, det(I− tσ2) = (1− t)(1− t2)and det(1− tσ3) = (1− t3) where σ2 = (1, 2) and σ3 = (1, 2, 3) .

105

106 CHAPTER 33. MOLIEN SERIES

Then

M(t) =1

6

(1

(1− t)3+

3

(1− t)(1− t2)+

2

1− t3

)=

1

(1− t)(1− t2)(1− t3)

33.4 References• David A. Cox, John B. Little, Donal O'Shea (2005), Using Algebraic Geometry, pp. 295–8

• Molien, Th. (1897). “Uber die Invarianten der linearen Substitutionsgruppen.”. Sitzungber. Konig. Preuss.Akad. Wiss. (J. Berl. Ber.) 52: 1152–1156. JFM 28.0115.01.

• Mukai, S. (2002). An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics 81.ISBN 978-0-521-80906-1.

Chapter 34

The Classical Groups

In Weyl’s wonderful and terrible1 book The Classical Groups [W] one may discern two main themes: first, the studyof the polynomial invariants for an arbitrary number of (contravariant or covariant) variables for a standard classicalgroup action; second, the isotypic decomposition of the full tensor algebra for such an action.1Most people who know the book feel the material in it is wonderful. Many also feel the presentation is terrible. (Theauthor is not among these latter.)Howe (1989, p.539)

In mathematics, The Classical Groups: Their Invariants and Representations is a book by Weyl (1939), whichdescribes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interestin invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s.Weyl (1939b) gave an informal talk about the topic of his book.

34.1 Contents

Chapter I defines invariants and other basic ideas and describes the relation to Felix Klein's Erlanger program ingeometry.Chapter II describes the invariants of the special and general linear group of a vector space V on the polynomials overa sum of copies of V and its dual. It uses the Capelli identity to find an explicit set of generators for the invariants.Chapter III studies the group ring of a finite group and its decomposition into a sum of matrix algebras.Chapter IV discusses Schur–Weyl duality between representations of the symmetric and general linear groups.Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal andsymplectic groups, showing that the ring of invariants is generated by the obvious ones.Chapter VII describes the Weyl character formula for the characters of representations of the classical groups.Chapter VIII on invariant theory proves Hilbert’s theorem that invariants of the special linear group are finitely gen-erated.Chapter IX and X give some supplements to the previous chapters.

34.2 References

• Howe, Roger (1988), "The classical groups and invariants of binary forms”, in Wells, R. O. Jr., The mathe-matical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math. 48, Providence, R.I.:American Mathematical Society, pp. 133–166, ISBN 978-0-8218-1482-6, MR 974333

• Howe, Roger (1989), “Remarks on classical invariant theory.”, Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539–570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR

107

108 CHAPTER 34. THE CLASSICAL GROUPS

2001418, MR 0986027

• Jacobson, Nathan (1940), “Book Review: The Classical Groups”, Bulletin of the American Mathematical So-ciety 46 (7): 592–595, doi:10.1090/S0002-9904-1940-07236-2, ISSN 0002-9904, MR 1564136

• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255

• Weyl, Hermann (1939), “Invariants”, Duke Mathematical Journal 5: 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030

34.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 109

34.3 Text and image sources, contributors, and licenses

34.3.1 Text• Bracket algebra Source: https://en.wikipedia.org/wiki/Bracket_algebra?oldid=646373285 Contributors: Michael Hardy, TakuyaMurata,

Rjwilmsi, Michael Slone, SmackBot, Cronholm144, Geometry guy, Nilradical, Sun Creator, Yobot, AnomieBOT, Citation bot, Citationbot 1 and Deltahedron

• Bracket ring Source: https://en.wikipedia.org/wiki/Bracket_ring?oldid=646373328Contributors: TakuyaMurata, Andrewman327, R.e.b.,TexasAndroid, Headbomb, David Eppstein, AnomieBOT, AdventurousSquirrel and Deltahedron

• Canonizant Source: https://en.wikipedia.org/wiki/Canonizant?oldid=665953899Contributors: Michael Hardy, R.e.b., Trappist themonk,K9re11 and Anonymous: 1

• Capelli’s identity Source: https://en.wikipedia.org/wiki/Capelli’s_identity?oldid=644260317 Contributors: Michael Hardy, CharlesMatthews, Alan Liefting, Giftlite, Bender235, Woohookitty, Rjwilmsi, R.e.b., Headbomb, David Eppstein, JL-Bot, Mild Bill Hiccup,Addbot, Yobot, Citation bot, Citation bot 1, Darij, Alexander Chervov, RjwilmsiBot, John of Reading, K9re11 and Anonymous: 9

• Catalecticant Source: https://en.wikipedia.org/wiki/Catalecticant?oldid=492018302 Contributors: Michael Hardy, Rjwilmsi, R.e.b.,Headbomb and Citation bot

• Cayley’sΩprocess Source: https://en.wikipedia.org/wiki/Cayley’s_%CE%A9_process?oldid=599029089Contributors: Michael Hardy,Gene Ward Smith, Anthony Appleyard, Salix alba, R.e.b., Beetstra, Headbomb, Magioladitis, JackSchmidt, Citation bot 1, AlexanderChervov and Anonymous: 1

• Chevalley–Iwahori–Nagata theorem Source: https://en.wikipedia.org/wiki/Chevalley%E2%80%93Iwahori%E2%80%93Nagata_theorem?oldid=648032681 Contributors: TakuyaMurata, R.e.b., Wavelength, TexasAndroid, Headbomb, David Eppstein, Yobot, BattyBot, Doc-torKubla and K9re11

• Chevalley–Shephard–Todd theorem Source: https://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem?oldid=635233489 Contributors: Michael Hardy, Giftlite, Gro-Tsen, Rjwilmsi, R.e.b., Closedmouth, Headbomb, Magioladitis, Geometryguy, Arcfrk, Yobot, Citation bot, Citation bot 1, Brad7777, K9re11 and Anonymous: 4

• Differential invariant Source: https://en.wikipedia.org/wiki/Differential_invariant?oldid=662863959Contributors: Michael Hardy, Camrn86,Henry Delforn (old), Charvest, Sławomir Biały, SporkBot and Helpful Pixie Bot

• Evectant Source: https://en.wikipedia.org/wiki/Evectant?oldid=649535632 Contributors: R.e.b., Metadox and K9re11• Geometric invariant theory Source: https://en.wikipedia.org/wiki/Geometric_invariant_theory?oldid=641869024Contributors: Takuya-

Murata, CharlesMatthews, Altenmann, Giftlite, Paul August, Gauge, BD2412, R.e.b., Hillman, Crasshopper, Arthur Rubin, Eigenlambda,JCSantos, RyanEberhart, RobHar, David Eppstein, LokiClock, Arcfrk, Stca74, JackSchmidt, Citation bot, Locobot, Citation bot 1,RobinK, Trappist the monk, Basemaze, D.Lazard, Brad7777, Jeremy112233, ChrisGualtieri, Enyokoyama, Mark viking and Anony-mous: 6

• Glossary of invariant theory Source: https://en.wikipedia.org/wiki/Glossary_of_invariant_theory?oldid=629724612Contributors: Rjwilmsi,Salix alba, R.e.b., Wavelength, Gilo1969, Trappist the monk and John of Reading

• Gram’s theorem Source: https://en.wikipedia.org/wiki/Gram’s_theorem?oldid=634856806 Contributors: Michael Hardy, Rjwilmsi,R.e.b., TexasAndroid, David Eppstein, Tassedethe, Trappist the monk, Jesse V. and K9re11

• Gröbner basis Source: https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis?oldid=667233351Contributors: AxelBoldt, Michael Hardy,AugPi, Charles Matthews, Tk~enwiki, Populus, Kevinatilusa, (:Julien:), Saforrest, Giftlite, Simplex~enwiki, 4pq1injbok, Paul August,Gauge, Liberatus, T.huckstep, Spoon!, Ntmatter, LutzL, Bricken, Gene Nygaard, Oleg Alexandrov, Joriki, Shreevatsa, Ruud Koot,Waldir, Rjwilmsi, Maxal, Jaraalbe, Hillman, Baccala@freesoft.org, NymphadoraTonks, Ibotty~enwiki, Arthur Rubin, Betacommand,Chris the speller, Chisophugis, Nbarth, Allansteel, FlyHigh, Lambiam, Archimerged, Rschwieb, Ylloh, CRGreathouse, CmdrObot,345Kai, Orangutan123, Pascal.Tesson, Hannes Eder, NBeale, Turgidson, Dricherby, .anacondabot, Liberio, David Eppstein, WATARU,JonMcLoone, Policron, Gfis, STBotD, Hqb, Rodhullandemu, DumZiBoT, T68492, Nicoguaro, Jaan Vajakas, Addbot, Lieven Smits,Williamaffleck, Miketeri, Luckas-bot, Monzloff, Zadneram, Sirius533, Pratikmoona, DR2006kl, Darij, RjwilmsiBot, Theodyl, ZéroBot,D.Lazard, M.Moreno-Maza, Frietjes, Helpful Pixie Bot, Nosuchforever, Aneishta, Mathprodigy20, Teammm, Jose Brox, FChyzak,T.Verron, Tudor987, Spidsen, Monkbot, Loraof and Anonymous: 63

• Haboush’s theorem Source: https://en.wikipedia.org/wiki/Haboush’s_theorem?oldid=665536134 Contributors: Charles Matthews,Giftlite, Jeff3000, Rjwilmsi, R.e.b., Sodin, SmackBot, BeteNoir, RobHar, David Eppstein, R'n'B, DavidCBryant, Addbot, Yobot, Citationbot, Ringspectrum, Citation bot 1, ZéroBot, Brad7777, ChrisGualtieri, Kernel-spaceman and Anonymous: 3

• Hall algebra Source: https://en.wikipedia.org/wiki/Hall_algebra?oldid=610954652 Contributors: Michael Hardy, Jitse Niesen, Giftlite,Linas, R.e.b., Headbomb, Turgidson, Vanish2, Arcfrk, Kilom691, Omnipaedista, Charvest, Stroppolo,Mark viking,MariaMonks, Vieque,Twocuteangel and Anonymous: 2

• Hermite reciprocity Source: https://en.wikipedia.org/wiki/Hermite_reciprocity?oldid=492034856Contributors: Michael Hardy, Giftlite,R.e.b. and RobHar

• Hilbert’s basis theorem Source: https://en.wikipedia.org/wiki/Hilbert’s_basis_theorem?oldid=673370489 Contributors: AxelBoldt,Bryan Derksen, Charles Matthews, MathMartin, Tobias Bergemann, Giftlite, Waltpohl, Zhen Lin, Brockert, Vivacissamamente, Guan-abot, Oleg Alexandrov, Guardian of Light, Kinu, R.e.b., Sodin, Jxr~enwiki, Ad4m, Hillman, Michael Slone, Gaius Cornelius, BeteNoir,Foxjwill, Lwassink, Tesseran, Rab V, ICPalm, Marek69, Asmeurer, Error792, STBot, Jobu0101, Arcfrk, SieBot, Randomblue, DumZ-iBoT, LaaknorBot, Lightbot, SettingMoon, Legobot, Luckas-bot, LilHelpa, Xqbot, FrescoBot, Jowa fan, Drusus 0, Anita5192, ClueBotNG, Brad7777, ChrisGualtieri, Brirush, Teddyktchan and Anonymous: 25

• Hilbert’s fourteenth problem Source: https://en.wikipedia.org/wiki/Hilbert’s_fourteenth_problem?oldid=626892914 Contributors:Zundark,Michael Hardy, Silverfish, CharlesMatthews, Gro-Tsen, RetiredUser2, Lejean2000, Natalya, GregorB, Rjwilmsi, R.e.b., Sango123,Mathbot, Polyade, MalafayaBot, BlackFingolfin, Mattbuck, Headbomb, Arcfrk, Mild Bill Hiccup, Addbot, DOI bot, GrouchoBot, Citationbot 1, Trappist the monk, BertSeghers and Anonymous: 1

110 CHAPTER 34. THE CLASSICAL GROUPS

• Hilbert’s syzygy theorem Source: https://en.wikipedia.org/wiki/Hilbert’s_syzygy_theorem?oldid=625047817 Contributors: CharlesMatthews, Hyacinth, Grendelkhan, Giftlite, Dratman, Zaphod Beeblebrox, Sodin, BeteNoir, Ylloh, Nick Number, David Eppstein,Myrizio, Policron, Arcfrk, Addbot, Brad7777 and Anonymous: 7

• Hilbert–Mumford criterion Source: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Mumford_criterion?oldid=640467779Contrib-utors: TakuyaMurata, R.e.b., TexasAndroid, David Eppstein, Yobot and K9re11

• Hodge bundle Source: https://en.wikipedia.org/wiki/Hodge_bundle?oldid=635372221 Contributors: Michael Hardy, Charles Matthews,Bender235, DESiegel, CBM, RobHar, Citation bot, Mono, Zvonkine, Helpful Pixie Bot and Brirush

• Invariant estimator Source: https://en.wikipedia.org/wiki/Invariant_estimator?oldid=661949375 Contributors: 3mta3, Zvika, Smack-Bot, Iridescent, CmdrObot, Headbomb,MungoKitsch,Melcombe, Jack-A-Roe, Rajagiryes, Yobot, 23dp, Kiefer.Wolfowitz, TheUtahrap-tor, BG19bot, Limit-theorem, SolidPhase and Anonymous: 2

• Invariant of a binary form Source: https://en.wikipedia.org/wiki/Invariant_of_a_binary_form?oldid=666111924Contributors: MichaelHardy, Charles Matthews, Rjwilmsi, R.e.b., JoergenB, LokiClock, Yobot, Citation bot, Helpful Pixie Bot, K9re11 and Anonymous: 4

• Invariant polynomial Source: https://en.wikipedia.org/wiki/Invariant_polynomial?oldid=607145492 Contributors: Michael Hardy, Sil-verfish, Charles Matthews, Fredrik, Nabla, Dialectric, David Eppstein, Addbot, Yobot, AnomieBOT, 777sms and Makecat-bot

• Invariant theory Source: https://en.wikipedia.org/wiki/Invariant_theory?oldid=661885946Contributors: Michael Hardy, CharlesMatthews,Michael Larsen, Rvollmert, Giftlite, Rgdboer, Ligulem, R.e.b., Ground Zero,Wavelength, Hillman, SmackBot, Polyade, Nbarth, Ligulem-bot, Headbomb, Sherbrooke, David Eppstein, Vegasprof, The enemies of god, Arcfrk, Alexbot, Addbot, Lightbot, PV=nRT, Javanbakht,Yobot, Citation bot, Omnipaedista, Dijkschneier, Darij, Jonesey95, TobeBot, Maxdlink, ClueBot NG, EdwardH, Mark L MacDonald,Stephan Alexander Spahn, Brirush, E E Ballew and Anonymous: 9

• Invariants of tensors Source: https://en.wikipedia.org/wiki/Invariants_of_tensors?oldid=647348578Contributors: Michael Hardy, CharlesMatthews, Sverdrup, BenFrantzDale, Pearle, Mpatel, Quiddity, Mathbot, Konrad West, Hillman, Alex Bakharev, Paul White, SmackBot,TimBentley, WhiteHatLurker, JRSpriggs, Myasuda, RDT2, Rbdupaix, Gcranston, Ebarbero, JackSchmidt, Ana109, Tomeasy, Ariel.amir,Arbitrarily0, Luckas-bot, Yobot, Xqbot, Cerniagigante, EmausBot, Saeedghalati, GeorgeRWeber, Ale.mlow, Elephantsofearth andAnony-mous: 14

• Kempf–Ness theorem Source: https://en.wikipedia.org/wiki/Kempf%E2%80%93Ness_theorem?oldid=647093192Contributors: Giftlite,R.e.b., David Eppstein, Yobot, AnomieBOT, Trappist the monk and K9re11

• Kostant polynomial Source: https://en.wikipedia.org/wiki/Kostant_polynomial?oldid=626940105Contributors: Michael Hardy, Giftlite,BD2412, Mathsci, David Eppstein, Veddharta, Makotoy, Yobot, Citation bot 1, EdoDodo, Chronulator, Helpful Pixie Bot, Citation-CleanerBot, Violapaul, Mark viking, The Disambiguator and Anonymous: 1

• Littlewood–Richardson rule Source: https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule?oldid=634739505 Con-tributors: Michael Hardy, Charles Matthews, Giftlite, Shreevatsa, Plrk, Magister Mathematicae, Rjwilmsi, R.e.b., Mhym, Makyen, Jam-baugh, Myasuda, Waldeck, JackSchmidt, KathrynLybarger, SchreiberBike, Marc van Leeuwen, WikHead, Zdaugherty, KarlJacobi, Ash-ton1983, Yobot, Citation bot 1, Darij, Jonesey95, Trappist the monk and Anonymous: 7

• Modular invariant theory Source: https://en.wikipedia.org/wiki/Modular_invariant_theory?oldid=655219327 Contributors: MichaelHardy, R.e.b., Headbomb, Addbot, Yobot, Suslindisambiguator, Luizpuodzius, ChrisGualtieri, Teddyktchan and Anonymous: 1

• Moduli space Source: https://en.wikipedia.org/wiki/Moduli_space?oldid=670426562 Contributors: AxelBoldt, Edward, Michael Hardy,TakuyaMurata, Charles Matthews, Michael Larsen, Rvollmert, JesseW, Tobias Bergemann, Giftlite, K igor k, AmarChandra, Lumidek,Gauge, Rgdboer, Keenan Pepper, Oleg Alexandrov, Rjwilmsi, Masnevets, Wavelength, Hillman, Michael Slone, Tong~enwiki, Crasshop-per, RDBury, Eskimbot, Gutworth, Nbarth, Lesnail, CRGreathouse, Myasuda, Thijs!bot, Dreamseeker02139, Vanish2, Jakob.scholbach,C quest000, The enemies of god, LokiClock, Hesam7, Stca74, WereSpielChequers, Mild Bill Hiccup, His Wikiness, 7&6=thirteen,Addbot, Yobot, AnomieBOT, False vacuum, Citation bot 1, GoingBatty, Stephan Spahn, Slawekb, Brandmeister, Helpful Pixie Bot,Enyokoyama, Mark viking, Samreid94, Monkbot, Allisaidwaswrong, KasparBot and Anonymous: 27

• Molien series Source: https://en.wikipedia.org/wiki/Molien_series?oldid=654572184 Contributors: Charles Matthews, Giftlite, MichaelSlone, Headbomb, JackSchmidt, Simplifix, Citation bot 1, Darij, RobinK, Helpful Pixie Bot and Anonymous: 1

• The Classical Groups Source: https://en.wikipedia.org/wiki/The_Classical_Groups?oldid=665137034 Contributors: Michael Hardy,Tobias Bergemann, JIP, Rjwilmsi, R.e.b., Epbr123, Headbomb, David Eppstein, Trappist the monk and Anonymous: 2

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