Grassmannians over rings and subpolygons

TL;DR

This paper explores special points on Grassmannians related to friezes with coefficients, using arithmetic matroids and cluster algebra specializations to connect geometric, algebraic, and combinatorial structures.

Contribution

It introduces a new perspective on Grassmannian points via friezes and subpolygons, linking cluster algebra specializations to hyperplane arrangements and Weyl groups.

Findings

Characterization of subpolygons in classic frieze patterns

Interpretation of hyperplane arrangements from cluster specializations

Representation of Weyl groups as generalized frieze patterns

Abstract

We investigate special points on the Grassmannian which correspond to friezes with coefficients in the case of rank two. Using representations of arithmetic matroids we obtain a theorem on subpolygons of specializations of the coordinate ring. As a special case we recover the characterization of subpolygons in classic frieze patterns. Moreover, we observe that specializing clusters of the coordinate ring of the Grassmannian to units yields representations that may be interpreted as arrangements of hyperplanes with notable properties. In particular, we get an interpretation of certain Weyl groups and groupoids as generalized frieze patterns.