Strong sign-coherency of certain symmetric polynomials, with application to cluster algebras

TL;DR

This paper introduces a family of polynomials linked to cluster algebras, conjecturing their symmetry and sign-coherent expansions across various symmetric polynomial bases, with implications for algebraic combinatorics.

Contribution

It defines new polynomials associated with cluster algebras and conjectures their symmetry and sign-coherence properties in multiple bases.

Findings

Polynomials are naturally linked to cluster algebras.

Conjecture: these polynomials are symmetric in variables.

Conjecture: their expansions are sign-coherent.

Abstract

For each positive integer n, we define a polynomial in the variables z_1,...,z_n with coefficients in the ring $\mathbb{Q}[q,t,r]$ of polynomial functions of three parameters q, t, r. These polynomials naturally arise in the context of cluster algebras. We conjecture that they are symmetric polynomials in z_1,...,z_n, and that their expansions in terms of monomial, Schur, complete homogeneous, elementary and power sum symmetric polynomials are sign-coherent.