Cluster multiplication in regular components via generalized Chebyshev polynomials

TL;DR

This paper introduces generalized Chebyshev polynomials to describe cluster characters in type A and infinite quivers, providing new combinatorial tools and explicit multiplication formulas for regular modules.

Contribution

It develops a multivariate generalization of Chebyshev polynomials and applies them to cluster algebras, offering a simplified combinatorial description and explicit formulas for regular modules.

Findings

Generalized Chebyshev polynomials arise in cluster characters for Dynkin and infinite quivers.

Provides explicit multiplication formulas for regular modules.

Simplifies the combinatorial understanding of type A cluster algebras.

Abstract

We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type $\mathbb A$ and representation-infinite quivers. This allows to obtain a simple combinatorial description of cluster algebras of type $\mathbb A$. We also provide explicit multiplication formulas for cluster characters associated to regular modules over the path algebra of any representation-infinite quiver.