Friezes satisfying higher SL$_k$determinants
Abstract
In this article, we construct SL$_k$friezes using Plücker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$spaces in $n$space via the Plücker embedding. When this cluster algebra is of finite type, the SL$_k$friezes are in bijection with the socalled mesh friezes of the corresponding Grassmannian cluster category. These are collections of positive integers on the ARquiver of the category with relations inherited from the mesh relations on the category. In these finite type cases, many of the SL$_k$friezes arise from specialising a cluster to 1. These are called unitary. We use IyamaYoshino reduction to analyse the nonunitary friezes. With this, we provide an explanation for all known friezes of this kind. An appendix by Cuntz and Plamondon proves that there are 868 friezes of type $E_6$.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.10562
 arXiv:
 arXiv:1810.10562
 Bibcode:
 2018arXiv181010562B
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Combinatorics;
 05E10;
 13F60;
 16G20;
 18D99;
 14M15
 EPrint:
 With an appendix by M. Cuntz and P.G. Plamondon