Séminaire Lotharingien de Combinatoire, 86B.52 (2022), 12 pp.
Esther Banaian, Sunita Chepuri, Elizabeth Kelley and
Sylvester W. Zhang
Rooted Clusters for Graph LP Algebras
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 \emph{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.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
The following versions are available: