University of Minnesota Combinatorics Seminar: 2020-2021

Date: 01/22/2021

Speaker: Dani Kaufman

Combinatorics of Affine Cluster Algebras
In joint work with Zachary Greenberg (UMD), we construct a natural finite quotient of the cluster complex of cluster algebras associated with affine Dynkin diagrams. We show that there is a simple and uniform formula which counts the number of clusters in these finite quotients, in a way generalizing the appearance of the Catalan numbers in reference to the finite cluster algebras. I will give a general overview of this construction for all affine types and give a proof of the cluster count in the affine A case using generating functions and triangulations of an annulus. If time permits, I will mention some generalizations of these ideas to cluster algebras associated with extended affine dynkin diagrams.

Combinatorics of Affine Cluster Algebras

In joint work with Zachary Greenberg (UMD), we construct a natural finite quotient of the cluster complex of cluster algebras associated with affine Dynkin diagrams. We show that there is a simple and uniform formula which counts the number of clusters in these finite quotients, in a way generalizing the appearance of the Catalan numbers in reference to the finite cluster algebras. I will give a general overview of this construction for all affine types and give a proof of the cluster count in the affine A case using generating functions and triangulations of an annulus. If time permits, I will mention some generalizations of these ideas to cluster algebras associated with extended affine dynkin diagrams.