Combinatorics Seminar: Combinatorics of X-variables in finite type cluster algebras

Seminar | April 9 | 12-1 p.m. | 939 Evans Hall

 Melissa Sherman-Bennett, UC Berkeley

 Department of Mathematics

A cluster algebra is a commutative ring determined by an initial "seed," which consists of A-variables, X-variables, and some additional data. Given a seed, one can produce new seeds via a combinatorial process called mutation. The cluster algebra is generated by the variables obtained from all possible sequences of mutations. In this talk, we will focus on cluster algebras of finite type, which are those with finitely many A- and X-variables. The classification of finite type cluster algebras, due to Fomin and Zelevinsky, coincides with the classification of reduced crystallographic root systems. For classical types, the combinatorics of the A-variables and their mutations are encoded by triangulations of marked surfaces associated to each type. In this talk, we will discuss how the X-variables fit into this combinatorial framework. Namely, we will show that in cluster algebras of classical types over the universal semifield, the X-variables are in bijection with the quadrilaterals (with a choice of diagonal) appearing in triangulations of the surface of the appropriate type. Using this bijection, we also give the number of X-variables in each type.

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