Explicit Grobner bases for Schubert slices

Matrix Schubert varieties are pullbacks to GL_n of Schubert varieties on the flag manifold described explicitly by certain determinantal equations on a generic matrix. Allen Knutson and Ezra Miller showed that a particular subset of the determinants form a Grobner basis and described the simplicial complex associated to the initial ideal in terms of pipe dreams.

We give analogous theorems for neighborhoods of points on Schubert varieties. These can also be explicitly described by determinantal equations on a matrix, and we show that the determinants form a Grobner basis and describe the associated simplicial complex. The proof is by induction on a vertex decomposition of this simplicial complex.