Schur Polynomials and The Yang-Baxter Equation

• Published in: Communications in mathematical physics 2011-12, Vol.308 (2), p.281-301
• Main Authors: Brubaker, Ben, Bump, Daniel, Friedberg, Solomon
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Complex Systems; Exact sciences and technology; Mathematical and Computational Physics; Mathematical methods in physics; Mathematical Physics; Other topics in mathematical methods in physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-011-1345-3

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map from to End , where V is a two-dimensional vector space such that if then R 12 ( g ) R 13 ( gh ) R 23 ( h ) = R 23 ( h ) R 13 ( gh ) R 12 ( g ). Here R i j denotes R applied to the i , j components of . The image of this map consists of matrices whose nonzero coefficients a 1 ,  a 2 ,  b 1 ,  b 2 ,  c 1 ,  c 2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a 1 a 2 + b 1 b 2 − c 1 c 2 = 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 s λ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.