Cluster structures in Schubert varieties in the Grassmannian

TL;DR

This paper demonstrates that the coordinate rings of open Schubert varieties in the Grassmannian can be described as cluster algebras, extending previous results and confirming a longstanding conjecture using combinatorial and algebraic techniques.

Contribution

It proves a folklore conjecture that the coordinate rings of Schubert varieties are cluster algebras, generalizing Scott's theorem and employing new combinatorial constructions.

Findings

Coordinate rings of Schubert varieties are cluster algebras.

Generalization to skew Schubert varieties using generalized plabic graphs.

Confirmation of a longstanding folklore conjecture in algebraic geometry.

Abstract

In this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is encoded using (target labelings of) Postnikov's plabic graphs. This result generalizes a theorem of Scott (Scott 2006) for the Grassmannian, and proves a folklore conjecture for Schubert varieties that has been believed by experts since (Scott 2006), though the statement was not formally written down until M\"uller-Speyer explicitly conjectured it (M\"uller-Speyer 2016). To prove this conjecture we use a result of Leclerc (Leclerc 2016), who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety can be identified with cluster algebras. Our proof also uses a construction of Karpman (Karpman 2016) to build plabic graphs associated to reduced expressions. We additionally generalize our result to the setting of skew Schubert varieties; the latter result uses generalized plabic graphs, i.e. plabic graphs whose boundary vertices need not be labeled in cyclic order.