Frieze matrices and infinite frieze patterns with coefficients

TL;DR

This paper introduces a new matrix framework for infinite frieze patterns with coefficients, providing a novel proof of the determinant properties related to these combinatorial structures.

Contribution

It presents a new type of matrix for infinite frieze patterns, offering an alternative proof of known determinant results and deepening the connection to cluster theory.

Findings

New matrix approach for infinite frieze patterns

Alternative proof of frieze determinant results

Enhanced understanding of frieze patterns in cluster theory

Abstract

Frieze patterns are combinatorial objects that are deeply related to cluster theory. Determinants of frieze patterns arise from triangular regions of the frieze, and they have been considered in previous works by Broline-Crowe-Isaacs, and by Baur-Marsh. In this article, we introduce a new type of matrix for any infinite frieze pattern. This approach allows us to give a new proof of the frieze determinant result given by Baur-Marsh.