Combinatorics of cluster structures in Schubert varieties

TL;DR

This paper provides a detailed combinatorial description of cluster structures in Schubert varieties of the Grassmannian using Postnikov's plabic graphs, extending previous work and proving conjectures with new constructions.

Contribution

It offers the first formal description of cluster structures in Schubert varieties, generalizing Scott's Grassmannian case and incorporating new combinatorial and algebraic techniques.

Findings

Explicit combinatorial description of cluster structures in Schubert varieties

Extension of Scott's Grassmannian description to more general cases

Construction of cluster seeds from reduced expressions and plabic graphs

Abstract

We give an explicit combinatorial description of cluster structures in Schubert varieties of the Grassmannian in terms of (target labelings of) Postnikov's plabic graphs. This description is a natural generalization of the description given by (Scott 2006) for the Grassmannian and has been believed by experts essentially since (Scott 2006), though the statement was not formally written down until (M\"uller-Speyer 2016). To prove this conjecture we use a result of (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 admit cluster structures. We also adapt a construction of (Karpman 2016) to build cluster seeds associated to reduced expressions. Further, we explicitly describe cluster structures in skew Schubert varieties using plabic graphs whose boundary vertices need not be labeled in cyclic order.