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

TL;DR

This paper explores the combinatorial properties of symmetric polynomials derived from integrable vertex models, establishing a quantum group deformation of Grothendieck polynomials and deriving their key properties.

Contribution

It introduces a new correspondence between wavefunctions of integrable models and symmetric polynomials, extending the theory of Grothendieck polynomials via quantum group deformations.

Findings

Established the correspondence between wavefunctions and symmetric polynomials.

Derived determinant pairing formulas for the symmetric polynomials.

Obtained branching formulas through analysis of boundary partition functions.

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 $U_q(sl_2)$ $R$-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.