Calculating entries of unitary ${\mathrm{SL}}_3$-friezes

Abstract

In this article, we consider tame ${\mathrm{SL}}_3$-frieze patterns that arise by specializing a cluster of Plücker variables in the coordinate ring of the Grassmannian $\mathcal{G}(3,n)$ to $1$. We show how to calculate arbitrary entries of such frieze patterns from the cluster in question. Let $\mathcal{F}$ be such a cluster. We study the set ${\mathcal{F}}_x$ of cluster variables in $\mathcal{F}$ that share a given index $x$ and derive a structure theorem for ${\mathcal{F}}_x$. These sets prove central to calculating the first and last non-trivial rows of the frieze pattern. After that, simple recursive formulas can be used to calculate all remaining entries.