Higher SL(k)-friezes
Sira Gratz
Abstract: Classical frieze patterns are combinatorial structures which relate back to Gauss’ Pentagramma Mirificum, and have been extensively studied by Conway and Coxeter in the 1970’s.
A classical frieze pattern is an array of numbers satisfying a local (2 × 2)- determinant rule. Conway and Coxeter gave a beautiful and natural classification of SL(2)-friezes via triangulations of polygons. One way to generalise the notion of a classical frieze pattern is to ask of such an array to satisfy a (k × k)-determinant rule instead, for k at least 2, leading to the notion of higher SL(k)-friezes. While the task of classifying classical friezes yields a very satisfying answer, higher SL(k)-friezes are not that well understood to date.
In this talk, we’ll discuss how one might start to fathom higher SL(k)-frieze patterns. The links between frieze patterns and cluster combinatorics encoded by triangulations of polygons in the k=2 case suggests a link to Grassmannian varieties under the Plücker embedding and the cluster algebra structure on their homogeneous coordinate rings. We find a way to exploit this relation for higher SL(k)-friezes and provide an easy way to generate SL(k)-friezes via Grassmannian combinatorics.
This talk is based on joint work with Baur, Faber, Serhiyenko and Todorov.
combinatoricscategory theoryrepresentation theory
Audience: researchers in the topic
( slides )
The TRAC Seminar - Théorie de Représentations et ses Applications et Connections
Organizers: | Thomas Brüstle*, Souheila Hassoun |
*contact for this listing |