Friezes, Strings and Cluster Variables

TL;DR

This paper establishes a connection between Laurent polynomials associated with walks in quivers and cluster characters of string modules over 2-Calabi-Yau tilted algebras, revealing a precise algebraic correspondence.

Contribution

It proves that Laurent polynomials from walks in quivers match cluster characters of string modules, up to a normalizing factor, for 2-Calabi-Yau tilted algebras.

Findings

Laurent polynomials coincide with cluster characters

Explicit normalizing monomial factor identified

Connection between walks in quivers and module invariants

Abstract

To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi-Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module, up to an explicit normalising monomial factor.