Exploiting Symmetries Alternating Sign Matrices and the Weyl Character Formulas David M. Bressoud Macalester College, St. Paul, MN Stanford University April 12, 2006
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
(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)
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
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.
Cauchy 1815 Any alternating function in divided by the Vandermonde determinant yields a symmetric function:
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 .
is the dimension of the representation Note that the symmetric group on n letters is the group of transformations of
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).
The dimension of the representation is
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
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
Weyl 1939 The Classical Groups: their invariants and representations: The Denominator Formulas
A different approach to the Vandermonde determinant formula
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 11 is the entry in that matrix, this uniquely defines the determinant for all square matrices. Carl Jacobi (1804–1851)
David Robbins (1942–2003)
Sum is over all alternating sign matrices, N(A) = # of –1’s
n 1 2 3 4 5 6 7 8 9 An 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 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
There is exactly one 1 in the first row 2 3 4 5 6 7 8 9 An 42 429 7436 218348 10850216 911835460 There is exactly one 1 in the first row
There is exactly one 1 in the first row 2 3 4 5 6 7 8 9 An 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 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+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+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 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: Conjecture 1:
Conjecture 1: Conjecture 2 (corollary of Conjecture 1):
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):
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):
Conjecture: (MRR, 1983)
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
Zeilberger announces a proof that # of ASM’s equals December, 1992 Zeilberger announces a proof that # of ASM’s equals Doron Zeilberger Rutgers University
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.
Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture:
1996 Kuperberg announces a simple proof “Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices Greg Kuperberg UC Davis
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 ).
H O H O H O H O H O H H H H H H
Horizontal 1 Vertical –1
southwest northwest southeast northeast
N = # of vertical I = inversion number = N + # of SW x2, y3
1980’s Anatoli Izergin Vladimir Korepin SUNY Stony Brook
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
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
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:
Kuperberg, 2001: proved formulas for counting some new six-vertex models:
Kuperberg, 2001: proved formulas for many symmetry classes of ASM’s and some new ones
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
Number of n n ASM’s is 3–n(n–1)/2 times the dimension of the irreducible representation of GL2n indexed by
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
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
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.