Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice

Abstract

We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in finite lattice. We introduce the $L$-operator related with the $U_q(s{l}_2)$ $R$-matrix, and construct the wavefunctions and their duals. We prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the Grothendieck polynomials. This is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions. As applications of the correspondence between the wavefunctions and symmetric polynomials, we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the $B$-operators.

34 pages, 12 figures

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Cited by in corpus (5)

• Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions
• Symmetric functions and wavefunctions of the six-vertex model by Izergin-Korepin analysis
• Izergin-Korepin analysis on the projected wavefunctions of the generalized free-fermion model
• Inhomogeneous spin $q$-Whittaker polynomials
• Scalar products of the elliptic Felderhof model and elliptic Cauchy formula