Calculating entries of unitary 𝑆𝐿_𝟹-friezes

authored by

Lucas Surmann

Abstract

In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Pl\"ucker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such friezes from the cluster in question. Let $ \mathscr{F} $ be such a cluster. We study the set $ \mathscr{F}_x $ of cluster variables in $ \mathscr{F} $ that share a given index $ x $ and derive a structure Theorem for $ \mathscr{F}_x $. These sets prove central to calculating the first and last non-trivial rows of the frieze. After that, simple recursive formulas can be used to calculate all remaining entries.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Journal of Combinatorial Algebra

ISSN

2415-6302

Publication date

11.03.2025

Publication status

E-pub ahead of print

Peer reviewed

Yes

Research Area (based on ÖFOS 2012)

Combinatorics, Algebra, Graph theory

Electronic version(s)

https://doi.org/10.4171/JCA/111 (Access: Open)
https://doi.org/10.48550/arXiv.2404.09811 (Access: Open)