Calculating entries of unitary $\mathrm{SL}_{3}$-friezes

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 \mathscr{G}(3,n) to 1 . We show how to calculate arbitrary entries of such frieze patterns 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 pattern. After that, simple recursive formulas can be used to calculate all remaining entries.