Friezes satisfying higher slk-determinants

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Karin Baur
  • Eleonore Faber
  • Sira Gratz
  • Khrystyna Serhiyenko
  • Gordana Todorov
  • Michael Cuntz
  • Pierre Guy Plamondon

External Research Organisations

  • University of Leeds
  • University of Glasgow
  • University of Kentucky
  • Northeastern University
  • Université Paris-Saclay
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Details

Original languageEnglish
Pages (from-to)29-68
Number of pages40
JournalAlgebra and Number Theory
Volume15
Issue number1
Publication statusPublished - 1 Mar 2021

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.

Keywords

    Cluster category, Frieze pattern, Grassmannian, Iyama–Yoshino reduction, Mesh frieze, Unitary frieze

ASJC Scopus subject areas

Cite this

Friezes satisfying higher slk-determinants. / Baur, Karin; Faber, Eleonore; Gratz, Sira et al.
In: Algebra and Number Theory, Vol. 15, No. 1, 01.03.2021, p. 29-68.

Research output: Contribution to journalArticleResearchpeer review

Baur, K, Faber, E, Gratz, S, Serhiyenko, K, Todorov, G, Cuntz, M & Plamondon, PG 2021, 'Friezes satisfying higher slk-determinants', Algebra and Number Theory, vol. 15, no. 1, pp. 29-68. https://doi.org/10.2140/ant.2021.15.29
Baur, K., Faber, E., Gratz, S., Serhiyenko, K., Todorov, G., Cuntz, M., & Plamondon, P. G. (2021). Friezes satisfying higher slk-determinants. Algebra and Number Theory, 15(1), 29-68. https://doi.org/10.2140/ant.2021.15.29
Baur K, Faber E, Gratz S, Serhiyenko K, Todorov G, Cuntz M et al. Friezes satisfying higher slk-determinants. Algebra and Number Theory. 2021 Mar 1;15(1):29-68. doi: 10.2140/ant.2021.15.29
Baur, Karin ; Faber, Eleonore ; Gratz, Sira et al. / Friezes satisfying higher slk-determinants. In: Algebra and Number Theory. 2021 ; Vol. 15, No. 1. pp. 29-68.
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AU - Gratz, Sira

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AU - Todorov, Gordana

AU - Cuntz, Michael

AU - Plamondon, Pierre Guy

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