Vol. 15, No. 1, 2021

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Friezes satisfying higher $\mathrm{SL}_k$-determinants

Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko and Gordana Todorov

Appendix: Michael Cuntz and Pierre-Guy Plamondon

Vol. 15 (2021), No. 1, 29–68

In this article, we construct SLk-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 SLk-friezes are in bijection with the so-called mesh friezes of the corresponding Grassmannian cluster category. These are collections of positive integers on the AR-quiver of the category with relations inherited from the mesh relations on the category. In these finite type cases, many of the SLk-friezes arise from specializing a cluster to 1. These are called unitary. We use Iyama–Yoshino reduction to analyze 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 E6.

frieze pattern, mesh frieze, unitary frieze, cluster category, Grassmannian, Iyama–Yoshino reduction
Mathematical Subject Classification 2010
Primary: 05E10
Secondary: 13F60, 14M15, 16G20, 18D99
Received: 15 January 2019
Revised: 23 April 2020
Accepted: 27 June 2020
Published: 1 March 2021
Karin Baur
School of Mathematics
University of Leeds
United Kingdom
Eleonore Faber
School of Mathematics
University of Leeds
United Kingdom
Sira Gratz
School of Mathematics and Statistics
University of Glasgow
United Kingdom
Khrystyna Serhiyenko
Department of Mathematics
University of Kentucky
Lexington, KY
United States
Gordana Todorov
Department of Mathematics
Northeastern University
Boston, MA
United States
Michael Cuntz
Leibniz Universität Hannover
Institut für Algebra, Zahlentheorie und diskrete Mathematik
Fakultät für Mathematik und Physik
Pierre-Guy Plamondon
Laboratoire de Mathématiques de Versailles
UVSQ, CNRS, Université Paris-Saclay