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Quantum Cluster Algebras and Quivers

SPEAKER: Dylan Rupel

TITLE: Quantum Cluster Algebras and Quivers

ABSTRACT A cluster algebra is a subalgebra of an ambient field of rational functions $mathcal{F}$. The generators of the cluster algebra, called cluster variables, are not given at the outset but are recursively defined by a process called mutation. Mutation begins with a seed, consisting of the initial cluster of algebraically independent generators of $mathcal{F}$ and an exchange matrix. The exchange matrix determines how subsequent cluster variables are obtained from the initial cluster. One of the first main theorems is the so called “Laurent phenomenon” which asserts that all cluster variables can be written as Laurent polynomials in the initial cluster variables. The explicit computation of cluster variables in terms of the initial cluster is given by the Caldero-Chapoton formula which describes cluster variables as certain generating functions whose coefficients are Euler characteristics of Grassmannians of subrepresentations in quiver representations. In this talk I will describe the quantum analog of this story.