Séminaire Lotharingien de Combinatoire, 80B.55 (2018), 12 pp.

Melissa Sherman-Bennett

Combinatorics of X-variables in Finite Type Cluster Algebras

Abstract. We compute the number of X-variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For non-exceptional types, these numbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal) appearing in triangulations of certain marked surfaces. We conjecture that similar results hold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding the structure of finite type cluster algebras of geometric type.


Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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