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Friezes satisfying
higher $\mathrm{SL}_k$-determinants
Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko and Gordana TodorovAppendix: Michael Cuntz and Pierre-Guy Plamondon |
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Vol. 15 (2021), No. 1, 29–68
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Abstract |
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In this article, we construct -friezes using Plücker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of -spaces in -space via the Plücker embedding. When this cluster algebra is of finite type, the -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 -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 . |
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Keywords
frieze pattern, mesh frieze, unitary frieze, cluster
category, Grassmannian, Iyama–Yoshino reduction
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Mathematical Subject Classification 2010
Primary: 05E10
Secondary: 13F60, 14M15, 16G20, 18D99
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Milestones
Received: 15 January 2019
Revised: 23 April 2020
Accepted: 27 June 2020
Published: 1 March 2021
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Authors |
| Karin Baur | |
| School of Mathematics University of Leeds Leeds United Kingdom |
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| Eleonore Faber | |
| School of Mathematics University of Leeds Leeds United Kingdom |
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| Sira Gratz | |
| School of Mathematics and
Statistics University of Glasgow Glasgow United Kingdom |
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| Khrystyna Serhiyenko | |
| Department of Mathematics University of Kentucky Lexington, KY United States |
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| Gordana Todorov | |
| Department of Mathematics Northeastern University Boston, MA United States |
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| Michael Cuntz | |
| Leibniz Universität Hannover Institut für Algebra, Zahlentheorie und diskrete Mathematik Fakultät für Mathematik und Physik Hannover Germany |
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| Pierre-Guy Plamondon | |
| Laboratoire de Mathématiques de
Versailles UVSQ, CNRS, Université Paris-Saclay Versailles France |
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