Friezes satisfying higher SL$_k$-determinants

TL;DR

This paper constructs and analyzes SL_k-friezes using cluster algebra techniques and Plücker coordinates, linking them to Grassmannian categories and classifying finite type cases with a comprehensive enumeration.

Contribution

It introduces a method to construct SL_k-friezes via cluster structures and classifies finite type friezes, including a complete enumeration for type E6.

Findings

Constructed SL_k-friezes using Plücker coordinates.

Established bijection between friezes and mesh friezes in finite types.

Enumerated 868 friezes of type E6.

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 $E_6$.