Exploiting Symmetries

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. 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 GLnindexed 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. 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 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?

24. n 1 2 3 4 5 6 7 8 9 An 1 2 7 42 429 7436 218348 10850216 911835460 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. 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

26. 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

27. 1 1 1 2 3 2 7 14 14 7 42 105 135 105 42 429 1287 2002 2002 1287 429

28. 1 1 1 2 3 2 7 14 14 7 42 105 135 105 42 429 1287 2002 2002 1287 429 + + +

29. 1 1 1 2 3 2 7 14 14 7 42 105 135 105 42 429 1287 2002 2002 1287 429 + + +

30. 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

31. 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

32. 2/2 2/33/2 2/45/54/2 2/57/99/75/2 2/69/1416/1614/96/2

33. 2 23 254 2795 2916146

34. Numerators: 1+1 1+11+2 1+12+31+3 1+13+43+61+4 1+14+56+104+101+5

35. 1+1 1+11+2 1+12+31+3 1+13+43+61+4 1+14+56+104+101+5 Numerators: Conjecture 1:

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

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

38. 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. 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 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

50. Horizontal  1 Vertical  –1