TY - JOUR

T1 - Schur Polynomials and The Yang-Baxter Equation

AU - Brubaker, Ben

AU - Bump, Daniel

AU - Friedberg, Solomon

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/12

Y1 - 2011/12

N2 - We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map {mapping} R(g) from GL(2,ℂ) × GL(1,ℂ) to End (V ⊗ V), where V is a two-dimensional vector space such that if g,h ε G then R12(g)R13(gh) R23(h) = R23(h) R13(gh)R12(g). Here Rij denotes R applied to the i, j components of V ⊗ V ⊗ V. The image of this map consists of matrices whose nonzero coefficients a1, a2, b1, b2, c1, c2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a1a2 + b1b2 - c1c2 = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial sλ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.

AB - We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map {mapping} R(g) from GL(2,ℂ) × GL(1,ℂ) to End (V ⊗ V), where V is a two-dimensional vector space such that if g,h ε G then R12(g)R13(gh) R23(h) = R23(h) R13(gh)R12(g). Here Rij denotes R applied to the i, j components of V ⊗ V ⊗ V. The image of this map consists of matrices whose nonzero coefficients a1, a2, b1, b2, c1, c2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a1a2 + b1b2 - c1c2 = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial sλ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.

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U2 - 10.1007/s00220-011-1345-3

DO - 10.1007/s00220-011-1345-3

M3 - Article

AN - SCOPUS:80355146886

VL - 308

SP - 281

EP - 301

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -