2009-09-11

Allen Knutson, Cornell University

Degenerations of Schubert varieties

I'll recall the definition of a Schubert variety X_w in a flag manifold G/B, and the Bott-Samelson(-Demazure-Hansen) resolutions thereof.
In a precise sense, Schubert varieties cannot degenerate inside G/B. So I will only degenerate affine patches on them, and prove the following:

1) Each patch degenerates to a union of coordinate spaces.

2) With respect to Bott-Samelson coordinates, the degeneration is controlled by a lexicographic Gr\"obner basis.

3) The combinatorics of the resulting Stanley-Reisner scheme is given by a "subword complex'', which is a shellable (even vertex-decomposable) ball.

This recovers a number of known results. If time permits I will discuss multiplicities on Schubert and Richardson varieties.
One crucial ingredient is a theorem about degenerations (independent of Lie theory) that I will prove in Wednesday's commutative algebra seminar.