Quantized Chebyshev polynomials and cluster characters with coefficients

TL;DR

This paper introduces quantized Chebyshev polynomials as deformations of classical polynomials, demonstrating their role in cluster algebras with principal coefficients and their interactions with bases, especially in affine types.

Contribution

It defines quantized Chebyshev polynomials and proves their relevance in cluster algebras with principal coefficients for specific quivers, extending previous work on classical polynomials.

Findings

Quantized Chebyshev polynomials arise in cluster algebras with principal coefficients.

They are connected to acyclic quivers of infinite representation types.

The polynomials interact with canonical bases in affine cluster algebras.

Abstract

We introduce quantized Chebyshev polynomials as deformations of generalized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomials arise in cluster algebras with principal coefficients associated to acyclic quivers of infinite representation types and equioriented Dynkin quivers of type $\mathbb A$. We also study their interactions with bases and especially canonically positive bases in affine cluster algebras.