Quantum frieze patterns in quantum cluster algebras of type A

TL;DR

This paper introduces a quantum version of Coxeter-Conway frieze patterns and demonstrates their role in representing quantum cluster variables in type A quantum cluster algebras, providing explicit polynomial formulas.

Contribution

It presents a novel quantisation of frieze patterns and establishes their connection to quantum cluster variables in type A, expanding the understanding of quantum cluster algebra structures.

Findings

Quantum frieze patterns realise quantum cluster variables.

Explicit polynomials are derived from the lower bound phenomenon.

The work applies to linearly oriented Dynkin quivers of type A.

Abstract

We introduce a quantisation of the Coxeter-Conway frieze patterns and prove that they realise quantum cluster variables in quantum cluster algebras associated with linearly oriented Dynkin quivers of type A. As an application, we obtain the explicit polynomials arising from the lower bound phenomenon in these quantum cluster algebras.