Colored five-vertex models and Lascoux polynomials and atoms

TL;DR

This paper introduces a new integrable colored five-vertex model that provides the first combinatorial interpretation of Lascoux polynomials and atoms, connecting statistical mechanics models with algebraic combinatorics.

Contribution

It constructs a novel integrable model whose partition function corresponds to Lascoux polynomials and atoms, and proves their combinatorial interpretations using set-valued tableaux.

Findings

First combinatorial interpretation of Lascoux polynomials and atoms.

Established connections between vertex models and algebraic combinatorics.

Proved conjectures relating to set-valued tableaux representations.

Abstract

We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai [arXiv:1305.3030] and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson [arXiv:1902.01795]. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first known combinatorial interpretation of a Lascoux polynomial and atom. Using this, we prove a conjectured combinatorial interpretation in terms of set-valued tableaux of a Lascoux polynomial and atom due to Pechenik and the second author [arXiv:1904.09674]. We also prove the combinatorial interpretation of the Lascoux atom using set-valued skyline tableaux of Monical [arXiv:1611.08777].