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arXiv · 1810.10562

Friezes satisfying higher SL$_k$-determinants

Karin Baur·Eleonore Faber·Sira Gratz·Khrystyna Serhiyenko·Gordana Todorov
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Abstract

In this article, we construct SL$_k$-friezes using Pl\"ucker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Pl\"ucker embedding. When this cluster algebra is of finite type, the SL$_k$-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 SL$_k$-friezes arise from specialising a cluster to 1. These are called unitary. We use Iyama-Yoshino reduction to analyse the non-unitary 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 $E_6$.

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Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko, Gordana Todorov. 2018-10-24. Friezes satisfying higher SL$_k$-determinants. https://doi.org/10.2140/ant.2021.15.29

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