### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 089, 30 pages      arXiv:2107.14785      https://doi.org/10.3842/SIGMA.2022.089

### Rooted Clusters for Graph LP Algebras

Esther Banaian a, Sunita Chepuri b, Elizabeth Kelley c and Sylvester W. Zhang d
a) Department of Mathematics, Aarhus University, 8000 Aarhus, Denmark
b) Department of Mathematics, Lafayette College, Easton, PA 18042, USA
c) Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA
d) School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received October 13, 2021, in final form November 17, 2022; Published online November 24, 2022

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.

Key words: Laurent phenomenon algebra; cluster algebra; graph LP algebra; $T$-path.

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