Cluster Monomials in Graph 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.
Keywords
Cite
@article{arxiv.2404.16153,
title = {Cluster Monomials in Graph Laurent Phenomenon Algebras},
author = {Guilherme Zeus Dantas e Moura and Ramanuja Charyulu Telekicherla Kandalam and Dora Woodruff},
journal= {arXiv preprint arXiv:2404.16153},
year = {2025}
}
Comments
22 pages, 7 figures. Expanded Section 5 on Nonnegativity for Trees, and other minor corrections