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David M. BressoudMacalester College, St. Paul, MNTalk given at University of FloridaOctober 29, 2004

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The Vandermonde determinantWeyls character formulaeAlternating sign matricesThe six-vertex model of statistical mechanicsOkadas work connecting ASMs and character formulae

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Cauchy 1815Memoir on functions whose values are equal but of opposite sign when two of their variables are interchanged(alternating functions)Augustin-Louis Cauchy (17891857)

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Cauchy 1815Memoir 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

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Cauchy 1815Memoir 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.

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

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Cauchy 1815Any alternating function in divided by the Vandermonde determinant yields a symmetric function: Called the Schur function. I.J. Schur (1917) recognized it as the character of the irreducible representation of GLn indexed by .Issai Schur (18751941)

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is the dimension of the representationNote that the symmetric group on n letters is the group of transformations of

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Weyl 1939 The Classical Groups: their invariants and representationsis the character of the irreducible representation, indexed by the partition , of the symplectic group (the subgoup of GL2n of isometries). Hermann Weyl (18851955)

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The dimension of the representation is

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

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

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

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Desnanot-Jacobi adjoint matrix thereom (Desnanot for n 6 in 1819, Jacobi for general case in 1833is 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 (18041851)

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David Robbins (19422003)

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

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How many n n alternating sign matrices?

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= 2 3 7= 3 11 13= 22 11 132= 22 132 17 19= 23 13 172 192= 22 5 172 193 23

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There is exactly one 1 in the first row

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There is exactly one 1 in the first row

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11 12 3 27 14 14 742 105 135 105 42429 1287 2002 2002 1287 429

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11 12 3 27 14 14 742 105 135 105 42429 1287 2002 2002 1287 429

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11 12 3 27 14 14 742 105 135 105 42429 1287 2002 2002 1287 429

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11 2/2 12 2/3 3 3/2 27 2/4 14 14 4/2 742 2/5 105 135 105 5/2 42429 2/6 1287 2002 2002 1287 6/2 429

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11 2/2 12 2/3 3 3/2 27 2/4 14 5/5 14 4/2 742 2/5 105 7/9 135 9/7 105 5/2 42429 2/6 1287 9/14 2002 16/16 2002 14/9 1287 6/2 429

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2/22/3 3/2 2/4 5/5 4/2 2/5 7/9 9/7 5/22/6 9/14 16/16 14/9 6/2

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22 3 2 5 4 2 7 9 52 9 16 14 6

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1+1 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:

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1+1 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 Conjecture 1:Numerators:

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

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

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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

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Conjecture:(MRR, 1983)

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Mills & Robbins (suggested by Richard Stanley) (1991)Symmetries of ASMsVertically symmetric ASMsHalf-turn symmetric ASMsQuarter-turn symmetric ASMs

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December, 1992Zeilberger announces a proof that # of ASMs equalsDoron ZeilbergerRutgers University

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December, 1992Zeilberger announces a proof that # of ASMs equals1995 all gaps removed, published as Proof of the Alternating Sign Matrix Conjecture, Elect. J. of Combinatorics, 1996.

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Zeilbergers proof is an 84-page tour de force, but it still left open the original conjecture:

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1996 Kuperberg announces a simple proofAnother proof of the alternating sign matrix conjecture, International Mathematics Research NoticesGreg KuperbergUC Davis

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Another proof of the alternating sign matrix conjecture, International Mathematics Research NoticesPhysicists have been studying ASMs for decades, only they call them square ice (aka the six-vertex model ).1996 Kuperberg announces a simple proof

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H O H O H O H O H O H H H H H HH O H O H O H O H O H H H H H HH O H O H O H O H O H H H H H HH O H O H O H O H O H H H H H HH O H O H O H O H O H

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Horizontal 1Vertical 1

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southwestnortheastnorthwestsoutheast

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N = # of verticalI = inversion number = N + # of SWx2, y3

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Anatoli IzerginVladimir KorepinSUNY Stony Brook1980s

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Proof:LHS is symmetric polynomial in xs and in ysDegree n 1 in x1By induction, LHS = RHS when x1 = y1Sufficient to show that RHS is symmetric polynomial in xs and in ys

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LHS is symmetric polynomial in xs and in ysDegree n 1 in x1By induction, LHS = RHS when x1 = y1Sufficient to show that RHS is symmetric polynomial in xs and in ys follows from Baxters triangle-to-triangle relationProof:Rodney J. BaxterAustralian National University

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1996Doron Zeilberger uses this determinant to prove the original conjectureProof of the refined alternating sign matrix conjecture, New York Journal of Mathematics

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

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

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Kuperberg, 2001: proved formulas for many symmetry classes of ASMs and some new ones

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Soichi Okada, Nagoya University1993, Okada finds the equivalent of the -determinant for the other Weyl Denominator Formulas.2004, Okada shows that the formulas for counting ASMs, including those subject to symmetry conditions, are simply the dimensions of certain irreducible representations, i.e. specializations of Weyl Character formulas.

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Number of n n ASMs is 3n(n1)/2 times the dimension of the irreducible representation of GL2n indexed by

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Number of (2n+1) (2n+1) vertically symmetric ASMs is 3n(n1) times the dimension of the irreducible representation of Sp4n indexed by

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NEW for 2004: Number of (4n+1) (4n+1) vertically and horizontally symmetric ASMs is 22n 3n(2n1) times

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Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture Cambridge University Press & MAA, 1999OKADA, Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups, arXiv:math.CO/0408234 v1 18 Aug 2004

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