Khrystyna Serhiyenko: Frieze patterns and representation theory

Описание: The goal of this talk is to explore the connections between various frieze patterns and representation theory of associative algebras. We begin with the classical Conway- Coxeter friezes over positive integers and their correspondence with Jacobian algebras of type A, where entries in the frieze count the number of submodules of indecompos- able representations. This can also be reinterpreted in terms of applying the Caldero- Chapoton map, providing a close connection to Fomin-Zelevinsky's cluster algebras. Extending these ideas beyond the classical case, we will also discuss higher dimen- sional friezes, called (tame) SLk friezes, as well as their relation to cluster algebras on coordinate rings of Grassmannians Gr(k,n) and their categorification. Furthermore, SLk friezes are a special type of SLk tilings, integer tilings of the plane satisfying the condition that every k x k square has determinant 1. We will present a characterization of SLk tilings in terms of pairs of bi-infinite sequences in Zk and discuss applications to duality and positivity.

Recording during the thematic meeting : «Frieze patterns in algebra, combinatorics and geometry» the May 12, 2025 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker : Guillaume Hennenfent

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