Weak Separation, Pure Domains and Cluster Distance

TL;DR

This paper investigates weakly separated collections related to fixed sets, establishing maximality conditions, deriving formulas for their size, and applying these results to cluster algebra structures on Grassmannians.

Contribution

It proves maximality of weakly separated collections under generic conditions and provides formulas for their sizes, advancing understanding of cluster distances in Grassmannian cluster algebras.

Findings

Maximal weakly separated collections are also maximal by size under generic conditions.

Derived a formula for the size of weakly separated collections based on fixed sets.

Calculated exact cluster and mutation distances for certain cluster variables.

Abstract

Following the proof of the purity conjecture for weakly separated collections, recent years have revealed a variety of wider examples of purity in different settings. In this paper we consider the collection $\mathcal A_{I,J}$ of sets that are weakly separated from two fixed sets $I$ and $J$. We show that all maximal by inclusion weakly separated collections $\mathcal W\subset\mathcal A_{I,J}$ are also maximal by size, provided that $I$ and $J$ are sufficiently "generic". We also give a simple formula for the cardinality of $\mathcal W$ in terms of $I$ and $J$. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.