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2021 Friezes satisfying higher $\mathrm{SL}_k$-determinants
Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko, Gordana Todorov
Algebra Number Theory 15(1): 29-68 (2021). DOI: 10.2140/ant.2021.15.29

Abstract

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.

Citation

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Karin Baur. Eleonore Faber. Sira Gratz. Khrystyna Serhiyenko. Gordana Todorov. "Friezes satisfying higher $\mathrm{SL}_k$-determinants." Algebra Number Theory 15 (1) 29 - 68, 2021. https://doi.org/10.2140/ant.2021.15.29

Information

Received: 15 January 2019; Revised: 23 April 2020; Accepted: 27 June 2020; Published: 2021
First available in Project Euclid: 17 March 2021

Digital Object Identifier: 10.2140/ant.2021.15.29

Subjects:
Primary: 05E10
Secondary: 13F60, 14M15, 16G20, 18D99

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.15 • No. 1 • 2021
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