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## Exploiting Symmetries

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**Exploiting Symmetries**Alternating Sign Matrices and the Weyl Character Formulas David M. Bressoud Macalester College, St. Paul, MN Talk given at University of Florida October 29, 2004**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**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 GLnindexed 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).**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**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)**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 very suspicious = 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**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**2/33/2 2/45/54/2 2/57/99/75/2 2/69/1416/1614/96/2**2**23 254 2795 2916146**Numerators:**1+1 1+11+2 1+12+31+3 1+13+43+61+4 1+14+56+104+101+5**1+1**1+11+2 1+12+31+3 1+13+43+61+4 1+14+56+104+101+5 Numerators: Conjecture 1:**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):**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**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 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 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