Kirillov's conjecture on Hecke-Grothendieck polynomials

TL;DR

This paper represents a broad class of polynomials, including Schubert and Grothendieck polynomials, as partition functions of solvable lattice models, proving positivity conjectures for Hecke-Grothendieck polynomials.

Contribution

It introduces algebraic methods from statistical mechanics to represent and analyze a new family of polynomials, confirming positivity conjectures for a key subfamily.

Findings

Proved positivity conjectures for Hecke-Grothendieck polynomials.

Represented polynomials as partition functions of solvable lattice models.

Identified instances of negative coefficients in the larger polynomial family.

Abstract

We use algebraic methods in statistical mechanics to represent a multi-parameter class of polynomials in several variables as partition functions of a new family of solvable lattice models. The class of polynomials, defined by A. N. Kirillov, is derived from the largest class of divided difference operators satisfying the braid relations of Cartan type $A$. It includes as specializations Schubert, Grothendieck, and dual-Grothendieck polynomials, among others. In particular, our results prove positivity conjectures of Kirillov for the subfamily of Hecke-Grothendieck polynomials, while the larger family is shown to exhibit rare instances of negative coefficients.