David M. BressoudMacalester College, St. Paul, MNTalk given at University of FloridaOctober 29, 2004
1. The Vandermonde determinant
2. Weyl’s character formulae
3. Alternating sign matrices
4. The six-vertex model of statistical mechanics
5. Okada’s work connecting ASM’s and character formulae
x1n 1 x2
n 1 L xnn 1
M M O Mx1 x2 L xn1 1 L 1
1 I xin i
i1
n
Sn
Cauchy 1815
“Memoir on functions whose values are equal but of opposite sign when two of their variables are interchanged”
(alternating functions) Augustin-Louis
Cauchy (1789–1857)
Cauchy 1815
“Memoir on functions whose values are equal but of opposite sign when two of their variables are interchanged”
(alternating functions)
This function is 0 when so it is divisible by
xi x j xi x j i j
x1n 1 x2
n 1 L xnn 1
M M O Mx1 x2 L xn1 1 L 1
1 I xin i
i1
n
Sn
Cauchy 1815
“Memoir on functions whose values are equal but of opposite sign when two of their variables are interchanged”
(alternating functions)
This function is 0 when so it is divisible by
xi x j xi x j i j
But both polynomials have same degree, so ratio is constant, = 1.
xi x j i j
x1n 1 x2
n 1 L xnn 1
M M O Mx1 x2 L xn1 1 L 1
1 I xin i
i1
n
Sn
Cauchy 1815
Any alternating function in divided by the Vandermonde determinant yields a symmetric function:
x1, x2 ,K , xn
x11 n 1 x2
1 n 1 L xn1 n 1
M M O Mx1
n 1 1 x2n 1 1 L xn
n 1 1
x1n x2
n L xnn
x1n 1 x2
n 1 L xnn 1
M M O Mx1
1 x21 L xn
1
x10 x2
0 L xn0
s x1, x2 ,K , xn
Cauchy 1815
Any alternating function in divided by the Vandermonde determinant yields a symmetric function:
x1, x2 ,K , xn
Called the Schur function. I.J. Schur (1917) recognized it as the character of the irreducible representation of GLn indexed by .
x11 n 1 x2
1 n 1 L xn1 n 1
M M O Mx1
n 1 1 x2n 1 1 L xn
n 1 1
x1n x2
n L xnn
x1n 1 x2
n 1 L xnn 1
M M O Mx1
1 x21 L xn
1
x10 x2
0 L xn0
s x1, x2 ,K , xn
Issai Schur (1875–1941)
s 1,1,K ,1 is the dimension of the representation
s 1,1,K ,1 r
rrAn 1
where n 1
2,n 3
2,K ,
1 n2
12
rrAn 1
,
1,2 ,K ,n ,An 1
ei e j 1 i j n ,
ei is the unit vector with 1 in the ith coordinate
Note that the symmetric group on n letters is the group of transformations of
An 1 ei e j 1 i j n
Weyl 1939 The Classical Groups: their invariants and representations
Sp2n ; rx
x11 n x1
1 n L xn1 n xn
1 n
M O Mx1
n 1 x1 n 1 L xn
n 1 xn n 1
x1n x1
n L xnn xn
n
M O Mx1
1 x1 1 L xn
1 xn 1
Sp2n ; rx is the character of the irreducible representation, indexed by the partition , of the symplectic group (the subgoup of GL2n of isometries).
Hermann Weyl (1885–1955)
Sp2n ;1r
r rrCn
where n,n 1,K ,1 12
rrCn
,
1,2 ,K ,n ,Cn
ei e j 1 i j n U 2ei 1 i n ,
ei is the unit vector with 1 in the ith coordinate
The dimension of the representation is
Weyl 1939 The Classical Groups: their invariants and representations
x1n x1
n L xnn xn
n
M O Mx1
1 x1 1 L xn
1 xn 1
x1x2L xn n
xi x j i j
is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at
1, x j 1 for 2 j n
Weyl 1939 The Classical Groups: their invariants and representations
x1n x1
n L xnn xn
n
M O Mx1
1 x1 1 L xn
1 xn 1
x1x2L xn n
xi x j i j
xi2 1
i xix j 1
i j
is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at
1, x j 1 for 2 j n
Weyl 1939 The Classical Groups: their invariants and representations: The Denominator Formulas
x1n 1
2 x1 n 1
2 L xnn 1
2 xn n 1
2
M O Mx1
12 x1
12 L xn
12 xn
12
x1x2L xn n 12
xi x j i j
xi 1 i
xix j 1 i j
x1n 1 x1
n1 L xnn 1 xn
n1
M O Mx1
0 x1 0 L xn
0 xn 0
x1x2L xn n 1
xi x j i j
2 xix j 1 i j
Desnanot-Jacobi adjoint matrix thereom (Desnanot for n ≤ 6 in 1819, Jacobi for general case in 1833M j
i is matrix M with row i and column j removed.
detM detM1
1 detM nn detM n
1 detM1n
detM1,n1,n
Given that the determinant of the empty matrix is 1 and the determinant of a 11 is the entry in that matrix, this uniquely defines the determinant for all square matrices.
Carl Jacobi (1804–1851)
detM detM1
1 detM nn detM n
1 detM1n
detM1,n1,n
det M detM1
1 detM nn detM n
1 detM1n
detM1,n1,n
det 1M detM
det a ji 1 i, j1
n ai a j
1i jn
David Robbins (1942–2003)
detM detM1
1 detM nn detM n
1 detM1n
detM1,n1,n
det M detM1
1 detM nn detM n
1 detM1n
detM1,n1,n
det
a bc d
ad bc
det
a b cd e fg h j
aej bdj afh 2 bfg cdh 3ceg
1 bde 1 fh
det
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4
a1,1a2,2a3,3a4,4
a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number
3 1 1 a1,2a2,1a2,2 1 a2,3a3,4a4,2 L
3 1 1 2a1,2a2,1a2,2
1 a2,3a3,2a3,3 1a3,4a4,3 L
det
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4
a1,1a2,2a3,3a4,4
a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number
3 1 1 a1,2a2,1a2,2 1 a2,3a3,4a4,2 L
3 1 1 2a1,2a2,1a2,2
1 a2,3a3,2a3,3 1a3,4a4,3 L
0 1 0 01 1 1 00 1 1 10 0 1 0
0 1 0 01 1 1 00 0 0 10 1 0 0
det
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4
a1,1a2,2a3,3a4,4
a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number
3 1 1 a1,2a2,1a2,2 1 a2,3a3,4a4,2 L
3 1 1 2a1,2a2,1a2,2
1 a2,3a3,2a3,3 1a3,4a4,3 L
0 1 0 01 1 1 00 1 1 10 0 1 0
0 1 0 01 1 1 00 0 0 10 1 0 0
det
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4
a1,1a2,2a3,3a4,4
a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number
3 1 1 a1,2a2,1a2,2 1 a2,3a3,4a4,2 L
3 1 1 2a1,2a2,1a2,2
1 a2,3a3,2a3,3 1a3,4a4,3 L
det xi, j Inv A
A ai , j 1 1 N A
xi, jai , j
i, j
Sum is over all alternating sign matrices, N(A) = # of –1’s
det
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4
a1,1a2,2a3,3a4,4
a1,2a2,1a3,3a4,4 a1,1a2,3a3,2a4,4 a1,1a2,2a3,4a4,3 L sums over other permutations inversion number
3 1 1 a1,2a2,1a2,2 1 a2,3a3,4a4,2 L
3 1 1 2a1,2a2,1a2,2
1 a2,3a3,2a3,3 1a3,4a4,3 L
det xi, j Inv A
A ai , j 1 1 N A
xi, jai , j
i, j
xi x j Inv A 1 1 N A
x jn i ai , j
i, j
AAn
1i jn
n
1
2
3
4
5
6
7
8
9
An
1
2
7
42
429
7436
218348
10850216
911835460
= 2 3 7= 3 11 13= 22 11 132
= 22 132 17 19= 23 13 172 192
= 22 5 172 193 23
How many n n alternating sign matrices?
n
1
2
3
4
5
6
7
8
9
An
1
2
7
42
429
7436
218348
10850216
911835460
= 2 3 7= 3 11 13= 22 11 132
= 22 132 17 19= 23 13 172 192
= 22 5 172 193 23
n
1
2
3
4
5
6
7
8
9
An
1
2
7
42
429
7436
218348
10850216
911835460
There is exactly one 1 in the first
row
n
1
2
3
4
5
6
7
8
9
An
1
1+1
2+3+2
7+14+14+7
42+105+…
There is exactly one 1 in the first
row
1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429
1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429
+ + +
1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429
+ + +
1 0 0 0 00 ? ? ? ?0 ? ? ? ?0 ? ? ? ?0 ? ? ? ?
1
1 2/2 1
2 2/3 3 3/2 2
7 2/4 14 14 4/2 7
42 2/5 105 135 105 5/2 42
429 2/6 1287 2002 2002 1287 6/2 429
1
1 2/2 1
2 2/3 3 3/2 2
7 2/4 14 5/5 14 4/2 7
42 2/5 105 7/9 135 9/7 105 5/2 42
429 2/6 1287 9/14 2002 16/16 2002 14/9 1287 6/2 429
2/2
2/3 3/2
2/4 5/5 4/2
2/5 7/9 9/7 5/2
2/6 9/14 16/16 14/9 6/2
2
2 3
2 5 4
2 7 9 5
2 9 16 14 6
1+1
1+1 1+2
1+1 2+3 1+3
1+1 3+4 3+6 1+4
1+1 4+5 6+10 4+10 1+5
Numerators:
1+1
1+1 1+2
1+1 2+3 1+3
1+1 3+4 3+6 1+4
1+1 4+5 6+10 4+10 1+5
Conjecture 1:
Numerators:
An,k
An,k1
n 2k 1
n 1k 1
n 2n k 1
n 1n k 1
Conjecture 1:
Conjecture 2 (corollary of Conjecture 1):
An,k
An,k1
n 2k 1
n 1k 1
n 2n k 1
n 1n k 1
An
3 j 1 !n j !j0
n 1
1!4!7!L 3n 2 !n!n 1 !L 2n 1 !
Conjecture 2 (corollary of Conjecture 1):
An
3 j 1 !n j !j0
n 1
1!4!7!L 3n 2 !n!n 1 !L 2n 1 !
Exactly the formula found by George Andrews for counting descending plane partitions.
George Andrews Penn State
Conjecture 2 (corollary of Conjecture 1):
An
3 j 1 !n j !j0
n 1
1!4!7!L 3n 2 !n!n 1 !L 2n 1 !
Exactly the formula found by George Andrews for counting descending plane partitions. In succeeding years, the connection would lead to many important results on plane partitions.
George Andrews Penn State
A n; x xN A
AAn
A 1; x 1,
A 2; x 2,A 3; x 6 x,
A 4; x 24 16x 2x2 ,
A 5; x 120 200x 94x2 14x3 x4 ,
A 6; x 720 2400x 2684x2 1284x3 310x4 36x5 2x6
A 7; x 5040 24900x 63308x2 66158x3 38390x4 13037x5
2660x6 328x7 26x8 x9
A n; x xN A
AAn
A 1; x 1,
A 2; x 2,A 3; x 6 x,
A 4; x 24 16x 2x2 ,
A 5; x 120 200x 94x2 14x3 x4 ,
A 6; x 720 2400x 2684x2 1284x3 310x4 36x5 2x6
A 7; x 5040 24900x 63308x2 66158x3 38390x4 13037x5
2660x6 328x7 26x8 x9
xi x j Inv A 1 1 N A
x jn i ai , j
i, j
AAn
1i jn
A n;0 n!
A n;1 An 3i 1 !n i !i0
n 1
A n;2 2n(n 1)/2
A n; x xN A
AAn
A 1; x 1,
A 2; x 2,A 3; x 6 x,
A 4; x 24 16x 2x2 ,
A 5; x 120 200x 94x2 14x3 x4 ,
A 6; x 720 2400x 2684x2 1284x3 310x4 36x5 2x6
A 7; x 5040 24900x 63308x2 66158x3 38390x4 13037x5
2660x6 328x7 26x8 x9
A n; 3 3n n 1
2n n 1 3 j i 1
3 j i 1i, jnj i odd
Conjecture:
(MRR, 1983)
A n;0 n!
A n;1 An 3i 1 !n i !i0
n 1
A n;2 2n(n 1)/2
Mills & Robbins (suggested by Richard Stanley) (1991)
Symmetries of ASM’s
A n 3 j 1 !n j !j0
n 1
AV 2n 1 3 n2 3 j i 1
j i 2n 11i, j2n12 j
A n
AHT 2n 3 n n 1 /2 3 j i 2j i ni, j
A n
AQT 4n AHT 2n A n 2
Vertically symmetric ASM’s
Half-turn symmetric ASM’sQuarter-turn symmetric ASM’s
December, 1992
Zeilberger announces a proof that # of ASM’s equals
3 j 1 !n j !j0
n 1
Doron Zeilberger
Rutgers University
December, 1992
Zeilberger announces a proof that # of ASM’s equals
3 j 1 !n j !j0
n 1
1995 all gaps removed, published as “Proof of the Alternating Sign Matrix Conjecture,” Elect. J. of Combinatorics, 1996.
Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture:
An,k
An,k1
n 2k 1
n 1k 1
n 2n k 1
n 1n k 1
1996 Kuperberg announces a simple proof
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices
Greg Kuperberg
UC Davis
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research NoticesPhysicists have been
studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).
1996 Kuperberg announces a simple proof
H O H O H O H O H O H
H H H H H
H O H O H O H O H O H
H H H H H
H O H O H O H O H O H
H H H H H
H O H O H O H O H O H
H H H H H
H O H O H O H O H O H
Horizontal 1
Vertical –1
southwest
northeast
northwest
southeast
N = # of verticalI = inversion number = N + # of SW
x2, y3
Anatoli IzerginVladimir Korepin
SUNY Stony Brook
1980’s
det1
xi y j axi y j
xi y j axi y j i, j1
nxi x j yi y j 1i jn
1 a 2N A an(n 1)/2 Inv A
AAn
xivert y j axi y j
SW, NE xi y j
NW, SE
Proof:LHS is symmetric polynomial in x’s and in y’s
Degree n – 1 in x1
By induction, LHS = RHS when x1 = y1
Sufficient to show that RHS is symmetric polynomial in x’s and in y’s
LHS is symmetric polynomial in x’s and in y’s
Degree n – 1 in x1
By induction, LHS = RHS when x1 = –y1
Sufficient to show that RHS is symmetric polynomial in x’s and in y’s — follows from Baxter’s triangle-to-triangle relation
Proof:
Rodney J. Baxter
Australian National University
a z 4 , xi z2 , yi 1
RHS z z 1 n n 1 z z 1 2N A
AAn
det1
xi y j axi y j
xi y j axi y j i, j1
nxi x j yi y j 1i jn
1 a 2N A an(n 1)/2 Inv A
AAn
xivert y j axi y j
SW, NE xi y j
NW, SE
det1
xi y j axi y j
xi y j axi y j i, j1
nxi x j yi y j 1i jn
1 a 2N A an(n 1)/2 Inv A
AAn
xivert y j axi y j
SW, NE xi y j
NW, SE
z e i / 3 : RHS 3 n n 1 /2 An ,
z e i / 4 : RHS 2 n n 1 /2 2N A
AAn ,
z e i /6 : RHS 1 n n 1 /2 3N A
AAn .
a z 4 , xi z2 , yi 1
RHS z z 1 n n 1 z z 1 2N A
AAn
1996
Doron Zeilberger uses this determinant to prove the original conjecture
“Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics
2001, Kuperberg uses the power of the triangle-to-triangle relation to prove some of the conjectured formulas:
AV 2n 1 3 n2 3 j i 1
j i 2n 1i, j2n12 j
A n
AHT 2n 3 n n 1 /2 3 j i 2j i ni, j
A n
AQT 4n AHT 2n A n 2
Kuperberg, 2001: proved formulas for counting some new six-vertex models:
AUU 2n 3 n2
22n 3 j i 2j i 2n 11i, j2n1
2 j
1 1 0 10 1 1 00 0 0 00 0 1 0
Kuperberg, 2001: proved formulas for many symmetry classes of ASM’s and some new ones
1 1 0 10 1 1 00 0 0 00 0 1 0
AUU 2n 3 n2
22n 3 j i 2j i 2n 11i, j2n1
2 j
Soichi Okada, Nagoya University
1993, Okada finds the equivalent of the -determinant for the other Weyl Denominator Formulas.
2004, Okada shows that the formulas for counting ASM’s, including those subject to symmetry conditions, are simply the dimensions of certain irreducible representations, i.e. specializations of Weyl Character formulas.
s 1,1,K ,1 r
rrA2n 1
3 j 1 2
3i 1 2
j i1i j2n
3 n(n 1)/2 s 1,1,K ,1 3i 1 !n i !i0
n 1
Number of n n ASM’s is 3–n(n–1)/2
times the dimension of the irreducible representation of GL2n indexed by n 1,n 1,n 2,n 2,K ,1,1,0,0
A2n 1 ei e j 1 i j 2n n 1
2,n 32,K , n 1
2
dim Sp4n C r
C rrC2n
6n 2
3i 12
3 j 1
2
4n 2 i j
1i j2n
3 j 1
2
3i 1
2
j i
3n 1 3i 1
2
2n 1 ii1
2n
Number of (2n+1) (2n+1) vertically symmetric ASM’s is 3–
n(n–1) times the dimension of the irreducible representation of Sp4n indexed by n 1,n 1,n 2,n 2,K ,1,1,0,0
C2n ei e j 1 i j 2n U 2ei 1 i 2n C
12
r rC2n
2n,2n 1,K ,1
NEW for 2004:
Number of (4n+1) (4n+1) vertically and horizontally symmetric ASM’s is 2–2n 3–n(2n–1)
times
n 1,n 1,n 2,n 2,K ,1,1,0,0 n 1
2,n 32,n 3
2,K , 32, 3
2, 12
dim Sp4n dim %O4n
C rC rrC2n
D rD rrD2n
C2n ei e j 1 i j 2n U 2ei 1 i 2n C 2n,2n 1,K ,1
D2n ei e j 1 i j 2n D 2n 1,2n 2,K ,1,0
Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture Cambridge University Press & MAA, 1999
OKADA, Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups, arXiv:math.CO/0408234 v1 18 Aug 2004