Combinatorial Formulas for Graph Laurent Phenomenon Algebras

Lam and Pylyavskyy's Laurent Phenomenon (LP) algebras are generalizations of Fomin-Zelevinsky's cluster algebras, in a way where exchange relations are relaxed to be any Laurent polynomial (rather than just binomials). LP algebras are known to satisfy Laurent positivity, but positivity remains open. We focus on a special class of LP algebras encoded by a graph (in this case, you may think of the graph playing the role of a "quiver"), and prove positivity for certain clusters when the graph is a tree. In particular, we give combinatorial formulas by means of hyper $T$-paths, which generalizes Schiffler's $T$-path formula for Type A surface cluster algebras. If time permits, I will also discuss current work-in-progress on constructing "hyper" snake graphs for these LP algebras. This talk is based on joint work with Esther Banaian, Sunita Chepuri and Elizabeth Kelly.