Schubert patches degenerate to subword complexes

Details Schubert patches degenerate to subword complexes Schubert patches degenerate to subword complexes

2008-01-27

arxiv.org/abs/0801.4114v1

Transformation Groups, Volume 13, Numbers 3-4 / December, 2008

Mathematics

Algebraic Geometry

11 pages

Scientific paper

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.

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