Colored five-vertex models and Demazure atoms

TL;DR

This paper constructs solvable lattice models with partition functions equal to Demazure atoms, linking combinatorics, algebra, and statistical mechanics, and introduces new algorithms for computing key elements in crystal bases.

Contribution

It introduces a novel colored five-vertex model whose partition functions are Demazure atoms, connecting lattice models with algebraic combinatorics.

Findings

Constructed solvable lattice models for Demazure atoms

Established a Yang-Baxter equation for the colored five-vertex model

Developed new algorithms for Lascoux-Schützenberger keys

Abstract

Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a biproduct, we construct Demazure atoms on Kashiwara's $\mathcal{B}_\infty$ crystal and give new algorithms for computing Lascoux-Sch\"utzenberger keys.