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.
|Number of pages||40|
|Journal||Algebra and Number Theory|
|State||Published - 2021|
Bibliographical noteFunding Information:
Baur was supported by FWF grants P 30549-N26 and W1230. She is supported by a Royal Society Wolfson Research Merit Award. Faber is a Marie Skłodowska-Curie fellow at the University of Leeds (funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 789580). Serhiyenko was supported by NSF Postdoctoral Fellowship MSPRF — 1502881. MSC2010: primary 05E10; secondary 13F60, 14M15, 16G20, 18D99. Keywords: frieze pattern, mesh frieze, unitary frieze, cluster category, Grassmannian, Iyama–Yoshino reduction.
© 2021 Mathematical Sciences Publishers.
- Cluster category
- Frieze pattern
- Iyama–Yoshino reduction
- Mesh frieze
- Unitary frieze
ASJC Scopus subject areas
- Algebra and Number Theory