1. Exploiting Symmetries
Alternating Sign Matrices
and the
Weyl Character Formulas
David M. Bressoud
Macalester College, St. Paul, MN
Stanford University
April 12, 2006

2. The Vandermonde determinant
Weyl’s character formulae
Alternating sign matrices
The six-vertex model of statistical mechanics
Okada’s work connecting ASM’s and character formulae

3. (alternating functions)
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)

4. 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

5. 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
But both polynomials have same degree, so ratio is constant, = 1.

6. Cauchy 1815
Any alternating function in divided by the Vandermonde determinant yields a symmetric function:

7. Cauchy 1815
Any alternating function in divided by the Vandermonde determinant yields a symmetric function:
Issai Schur (1875–1941)
Called the Schur function. I.J. Schur (1917) recognized it as the character of the irreducible representation of GLn indexed by .

8. is the dimension of the representation
Note that the symmetric group on n letters is the group of transformations of

9. Weyl 1939 The Classical Groups: their invariants and representations
Hermann Weyl (1885–1955)
is the character of the irreducible representation, indexed by the partition , of the symplectic group (the subgoup of GL2n of isometries).

10. The dimension of the representation is

11. Weyl 1939 The Classical Groups: their invariants and representations
is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at

12. Weyl 1939 The Classical Groups: their invariants and representations
is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at

13. Weyl 1939 The Classical Groups: their invariants and representations: The Denominator Formulas

14. A different approach to
the Vandermonde
determinant formula

15. is matrix M with row i and column j removed.
Desnanot-Jacobi adjoint matrix thereom (Desnanot for n ≤ 6 in 1819, Jacobi for general case in 1833
is matrix M with row i and column j removed.
Given that the determinant of the empty matrix is 1 and the determinant of a 11 is the entry in that matrix, this uniquely defines the determinant for all square matrices.
Carl Jacobi (1804–1851)

16. David Robbins (1942–2003)

21. Sum is over all alternating sign matrices, N(A) = # of –1’s

23. n 1 2 3 4 5 6 7 8 9 An 42
429
7436
218348
= 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?

24. n 1 2 3 4 5 6 7 8 9 An 42
429
7436
218348
very suspicious
= 2  3  7
= 3  11  13
= 22  11  132
= 22  132  17  19
= 23  13  172  192
= 22  5  172  193  23

25. There is exactly one 1 in the first row2 3 4 5 6 7 8 9 An 42
429
7436
218348
There is exactly one 1 in the first row

26. There is exactly one 1 in the first row2 3 4 5 6 7 8 9 An
1+1
2+3+2 …
There is exactly one 1 in the first row

27. 1

28. 1 + + +

29. 1 + + +

30. 1
1 2/2 1
2 2/ /2 2
7 2/ /2 7
42 2/ /2 42
429 2/ /2 429

31. 1
1 2/2 1
2 2/ /2 2
7 2/ / /2 7
42 2/ / / /2 42
429 2/ / / / /2 429

32. 2/2
2/ /2
2/ / /2
2/ / / /2
2/ / / / /2

33. 2

34. 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:

35. Numerators: 1+1 1+1 1+2 1+1 2+3 1+3 1+1 3+4 3+6 1+4
Numerators:
Conjecture 1:

36. Conjecture 1:
Conjecture 2 (corollary of Conjecture 1):

37. Conjecture 2 (corollary of Conjecture 1):
Exactly the formula found by George Andrews for counting descending plane partitions.
George Andrews Penn State
Conjecture 2 (corollary of Conjecture 1):

38. Conjecture 2 (corollary of Conjecture 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
Conjecture 2 (corollary of Conjecture 1):

41. Conjecture:
(MRR, 1983)

42. Mills & Robbins (suggested by Richard Stanley) (1991)
Symmetries of ASM’s
Vertically symmetric ASM’s
Half-turn symmetric ASM’s
Quarter-turn symmetric ASM’s

43. Zeilberger announces a proof that # of ASM’s equals
December, 1992
Zeilberger announces a proof that # of ASM’s equals
Doron Zeilberger
Rutgers University

44. December, 1992
Zeilberger announces a proof that # of ASM’s equals
1995 all gaps removed, published as “Proof of the Alternating Sign Matrix Conjecture,” Elect. J. of Combinatorics, 1996.

45. Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture:

46. 1996 Kuperberg announces a simple proof
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices
Greg Kuperberg
UC Davis

47. 1996 Kuperberg announces a simple proof
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices
Greg Kuperberg
UC Davis
Physicists have been studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).

48. H O H O H O H O H O H
H H H H H

50. Horizontal  1
Vertical  –1

51. southwest
northwest
southeast
northeast

52. N = # of vertical
I = inversion
number
= N + # of SW
x2, y3

53. 1980’s
Anatoli Izergin
Vladimir Korepin
SUNY Stony Brook

54. 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

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

58. 1996
Doron Zeilberger uses this determinant to prove the original conjecture
“Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

59. 2001, Kuperberg uses the power of the triangle-to-triangle relation to prove some of the conjectured formulas:

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

61. Kuperberg, 2001: proved formulas for many symmetry classes of ASM’s and some new ones

62. 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.
Soichi Okada, Nagoya University

63. Number of n  n ASM’s is 3–n(n–1)/2 times the dimension of the irreducible representation of GL2n indexed by

64. 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

65. NEW for 2006 (preprint 2004):
Number of (4n+1)  (4n+1) vertically and horizontally symmetric ASM’s is 2–2n 3–n(2n–1) times

66. 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, J. Algebraic Combinatorics, 23 (2006), 43–69.