Developments in Alternating Sign

Matrices David Bressoud Macalester College St. Paul, MN Rutgers University

New Brunswick, NJ October 14, 2010

PowerPointavailableatwww.macalester.edu/~bressoud/talks

MAA

1.  Developments leading to Doron’s proof of the refined alternating sign matrix conjecture.

2.  Counting symmetry classes of ASMs

3.  Fully packed loop configurations

4.  Fischer’s proof of the refined alternating sign matrix conjecture

5.  Fruits of an undergraduate’s exploration.

DavidRobbins(1942–2003)

Alternating Sign Matrices

Kuperberg’s representation

Monotone Triangle Strict increase across rows, weak increase along diagonals

12345

1234 1235 1245 1345 2345

12 13 14 15 23 24 25 34 35 45

1 2 3 4 5

3 2 2 4 3 2 5 4 3 2

123 124 125 134 135 145 234 235 245 345

x2 4

12345

1234 1235 1245 1345 2345

12 13 14 15 23 24 25 34 35 45

1 2 3 4 5

3 2 2 4 3 2 5 4 3 2

14 7 23 14 26 23 7 14 7 123 124 125 134 135 145 234 235 245 345

14

x x x 1 4 5

12345

1234 1235 1245 1345 2345

12 13 14 15 23 24 25 34 35 45

1 2 3 4 5

3 2

42 105 135 105 42

429

2 4 3 2 5 4 3 2

14 7 23 14 26 23 7 14 7 123 124 125 134 135 145 234 235 245 345

14

1

12/21

22/333/22

72/4145/5144/27

422/51057/91359/71055/242

4292/612879/14200216/16200214/912876/2429

An,k = # of n×n alternating sign matrices with 1 in row n, column k.

1

12/21

22/3 33/22

72/4 145/5 144/27

422/51057/91359/71055/242

4292/612879/14 200216/16 200214/912876/2429

Conjecture:

Conjecture1:

Conjecture2(corollaryofConjecture1):

1992: George Andrews proves Robbins conjecture that the number of totally symmetric, self-complementary plane partitions in an n×n×n box is given by

Z, Proof of the Alternating Sign Matrix Conjecture, Elect. J. of Combin., 1996.

1996 Kuperberg announces a simple proof

“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices Greg Kuperberg

UC Davis

Physicists had been studying ASM’s for decades, only they called them the six-vertex model.

Horizontal = 1

Vertical = –1

southwest

northeast

northwest

southeast

N = # of vertical vertices (the number of –1’s)

I = inversion number = N + # of SW

N = 2, I = 5

(x2, y3)

Anatoli Izergin Vladimir Korepin

SUNY Stony Brook

1980’s

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 y’s

Rodney J. Baxter

Australian National University

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 — This follows from Baxter’s triangle-to-triangle relation.

1996

Doron used this determinant to prove the original conjecture

“Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture

Cambridge University Press & MAA, 1999

2001, Kuperberg uses the power of the triangle-to-triangle relation to prove some of formulas conjectured by Robbins:

Vertically symmetric ASMs

1800 rotationally symmetric ASMs

900 rotationally symmetric ASMs

Kuperberg, 2001: proved formulas for counting some new six-vertex models:

2004, Soichi Okada observes that the number of n × n ASM’s is 3–n(n–1)/2 times the dimension of the irreducible representation of GL2n indexed by

A2n−1+ = ei − ej 1 ≤ i < j ≤ 2n{ }ρ = n − 12,n −

32,…,−n + 12( )

Okada proves the formula for vertically and horizontally symmetric ASMs:

Okada also proves the comparable formula for AVH (4n–1)

2006, Razumov and Stroganov prove the conjectured formula for quarter-turn symmetric ASMs of odd order:

There is one remaining unproven conjecture: The number of ASMs of odd order that are symmetric across both diagonals is

Fully packed loop configuration

Fully packed loop configuration

1

2

3 4 5

6

7

8 9 10

Each ASM defines a pairing of the vertices, π.

Ben Wieland, 2000, proved that the number of ASMs that correspond to a given pairing is unchanged if we rotate the labels.

1

2

3 4 5

6

7

8 9 10

Wieland’s theorem is a special case of the Razumov-Stroganoff conjecture:

The number of ASMs associated to a given pairing is a “component of a ground-state wavefunction.”

1

2

3 4 5

6

7

8 9 10

The Razumov-Stroganov conjecture was proven this past spring by physicists Luigi Cantini and Andrea Sportiello.

12 13 14 15 23 24 25 34 35 45

1 2 3 4 5

3 2 2 4 3 2 5 4 3 2

x2 4

2006, Ilse Fischer observes that the number of monotone triangles with bottom row k1, k2 is k2 – k1 + 1.

The number with bottom row k1, k2, k3 is

A(n;k1,…,kn) = the number of monotone triangles with bottom row k1,…,kn.

A(n;k1,…,kn) is a polynomial of total degree n(n–1)/2 and degree n–1 in each of the ki.

2007, Fischer provides a new proof of the refined ASM conjecture (the value of An,i).

She uses her characterization of A(n;k1,…,kn) to prove that

(An,1,…, An,n) is an eigenvector with eigenvalue 1. Verify that the dimension of the eigenspace is at most one and that the conjectured values satisfy this identify. Check constant.

1

12/21

22/333/22

72/4145/5144/27

422/51057/91359/71055/242

4292/612879/14200216/16200214/912876/2429

2008, Carleton undergraduate Nathan Williams looks at the alternating sums of the rows.

1

0

1

0

9

1: 1 3: 1 5: 9 7: 676 9: 417 316 11: 21054 33225 13: 8657 65116 22500 15: 28972 58363 89801 95600 17: 7 88319 29114 31396 91797 40176 19: 1742 93613 18275 76565 60875 92718 01924

AV(2n+1)2 is the number of ASMs of odd order with no 0’s on the central column, ACC(2n+1).

Williams asks: Is there a bijective proof ?

PowerPointavailableatwww.macalester.edu/~bressoud/talks