Cluster Monomials in Graph Laurent Phenomenon Algebras

TL;DR

This paper proves that cluster monomials form a linear basis in graph Laurent phenomenon algebras and shows nonnegativity of coefficients for bidirected trees, extending key cluster algebra results.

Contribution

It establishes the linear basis property for cluster monomials in graph Laurent phenomenon algebras and proves nonnegativity of coefficients for bidirected trees.

Findings

Cluster monomials form a linear basis in graph Laurent phenomenon algebras.

Coefficients of monomial expansions are nonnegative for bidirected trees.

Extends known cluster algebra results to a broader class of Laurent phenomenon algebras.

Abstract

Laurent phenomenon algebras, first introduced by Lam and Pylyavskyy, are a generalization of cluster algebras that still possess many salient features of cluster algebras. Graph Laurent phenomenon algebras, defined by Lam and Pylyavskyy, are a subclass of Laurent phenomenon algebras whose structure is given by the data of a directed graph. In this paper, we prove that the cluster monomials of a graph Laurent phenomenon algebra form a linear basis, as conjectured by Lam and Pylyavskyy and analogous to a result for cluster algebras by Caldero and Keller. We also prove that, if the graph is a bidirected tree, the coefficients of the expansion of any monomial in terms of cluster monomials are nonnegative.