Manchester Algebra Seminar - Sira Gratz - SL(k)-friezes
|Dates:||16 June 2020|
|Times:||14:00 - 15:00|
|What is it:||Seminar|
|Organiser:||Department of Mathematics|
|Who is it for:||University staff, Current University students|
Speaker: Sira Gratz - University of Glasgow
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 x 2)-determinant rule. Conway and Coxeter gave a beautiful and natural classification of SL(2)-friezes via triangulations of polygons. This same combinatorics occurs in the study of cluster algebras, and has revived interest in the subject. From this point of view, a natural way to generalise the notion of a classical frieze pattern is to ask of such an array to satisfy a (k x k)-determinant rule instead, for k bigger than 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 SL(2)-friezes and triangulations of polygons suggests a link to Grassmannian varieties under the Plücker embedding. We find a way to exploit this relation for higher SL(k)-friezes, and provide an easy way to generate a number of SL(k)-friezes via Grassmannian combinatorics, and suggest some ideas towards a complete classification using the theory of cluster algebras.
This talk is based on joint work with Baur, Faber, Serhiyenko and Todorov.
Travel and Contact Information
Zoom - online
Alan Turing Building