On Leclerc's conjectural cluster structures for open Richardson varieties

TL;DR

This paper confirms that Ménard's explicit seeds produce upper cluster algebra structures on open Richardson varieties, advancing the understanding of cluster structures in algebraic geometry.

Contribution

It demonstrates that Ménard's seeds indeed generate upper cluster algebra structures, clarifying the conjectural framework proposed by Leclerc.

Findings

Ménard's seeds yield upper cluster algebra structures

Discussion of remaining problems for full cluster algebra proof

Advancement in understanding cluster structures on Richardson varieties

Abstract

In 2016, Leclerc constructed conjectural cluster structures on open Richardson varieties using representations of preprojective algebras. A variant with more explicit seeds was obtained by M\'enard in his thesis. We show that M\'enard's seeds do yield *upper* cluster algebra structures on open Richardson varieties and discuss the problems that remain in order to prove that they are cluster algebra structures.