The Exchange Graphs of Weakly Separated Collections

TL;DR

This paper explores the structure of exchange graphs related to weakly separated collections in the context of cluster algebras and Grassmannians, establishing isomorphisms and characterizations for specific cases.

Contribution

It proves an isomorphism between exchange graphs and a class of -constant graphs, extending previous characterizations to four cases and characterizing graphs for cycles and trees.

Findings

Established an isomorphism between exchange graphs and -constant graphs.

Extended characterizations to the smallest four cases of these graphs.

Fully characterized graphs in the cases of cycles and trees.

Abstract

Weakly separated collections arise in the cluster algebra derived from the Pl\"ucker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace $\mathcal{I}$ and proved the connectivity of every exchange graph. Oh and Speyer later introduced a generalization of exchange graphs that we call $\mathcal{C}$-constant graphs. They characterized these graphs in the smallest two cases. We prove an isomorphism between exchange graphs and a certain class of $\mathcal{C}$-constant graphs. We use this to extend Oh and Speyer's characterization of these graphs to the smallest four cases, and we present a conjecture on a bound on the maximal order of these graphs. In addition, we fully characterize certain classes of these graphs in the special cases of cycles and trees.