Calculating entries of unitary $S{L}_3$-friezes

In this article we consider tame $S{L}_3$-friezes 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 friezes 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. After that, simple recursive formulas can be used to calculate all remaining entries.