Calculating entries of unitary $SL_3$-friezes

verfasst von

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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Preprint

Publikationsdatum

15.04.2024

Publikationsstatus

Elektronisch veröffentlicht (E-Pub)

Fachgebiet (basierend auf ÖFOS 2012)

Kombinatorik, Algebra, Graphentheorie