Schubert patches degenerate to subword complexes

TL;DR

This paper demonstrates how Schubert patches can be degenerated into subword complexes, providing new proofs for formulas related to restrictions of equivariant Schubert classes using combinatorial and geometric methods.

Contribution

It introduces an inductive degeneration of Schubert patches to Stanley-Reisner schemes with subword complexes, extending known results to new types and providing alternative proofs.

Findings

Schubert patches degenerate to Stanley-Reisner schemes with subword complexes

The underlying simplicial complex is homeomorphic to a ball

Provides new proofs for classical formulas in Schubert calculus

Abstract

We study the intersections of general Schubert varieties X_w with permuted big cells, and give an inductive degeneration of each such "Schubert patch" to a Stanley-Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley-Reisner scheme is a subword complex, as introduced for slightly different purposes in [Knutson-Miller 2004], and is homeomorphic to a ball. This gives a new proof of the Andersen-Jantzen-Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points.