Rooted Clusters for Graph LP Algebras

TL;DR

This paper introduces rooted clusters for graph LP algebras derived from trees, proving positivity through explicit formulas and combinatorial interpretations, advancing understanding of cluster variables in this algebraic framework.

Contribution

It defines rooted clusters for tree-based graph LP algebras and proves their positivity with explicit formulas and combinatorial models.

Findings

Proves positivity for rooted clusters in tree graph LP algebras.

Provides explicit formulas for cluster variables.

Offers a combinatorial interpretation using generalized T-paths.

Abstract

LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called rooted clusters. We prove positivity for these clusters by giving explicit formulas for each cluster variable. We also give a combinatorial interpretation for these expansions using a generalization of $T$-paths.