Combinatorics Seminar: Frieze patterns

Seminar | November 13 | 12:10-1 p.m. | 939 Evans Hall | Note change in date

 Khrystyna Serhiyenko, UC Berkeley

 Department of Mathematics

A frieze is a lattice of shifted rows of positive integers satisfying a diamond rule: the determinant of every 2 by 2 matrix formed by the neighboring entries is 1. Friezes were first studied by Conway and Coxeter in 1970's, but they gained fresh interest in the last decade in relation to cluster algebras. In particular, there exists a bijection between friezes and cluster algebras of type A. We introduce mutations of friezes, that are compatible with cluster mutations, and describe the resulting entries using combinatorics of quiver representations. We will also discuss an important generalization of the classical friezes, called $sl_k$ friezes, where the determinant of every k by k matrix is 1. In a similar manner, we investigate how $sl_k$ friezes can be obtained from cluster algebras associated to Grassmannians Gr(k,n). This is joint work with K. Baur, E. Faber, S. Gratz, and G. Todorov.