Singular Locus of Schubert Varieties

The singular locus of a Schubert variety \(X_v\) is determined by its smoothness at the \(T\)-fixed points indexed by permutations \(u\) less than \(v\) under the Bruhat order. We construct an affine open neighborhood around each \(u\) in \(X_v\), decompose it into a product of an affine variety with some affine space. We define a combinatorial relation on pairs of permutations called interval pattern embedding, generalizing the usual notion of pattern embedding. Finally, we show how the open neighborhoods associated with different pairs of \(u<v\) are related when interval embedding occurs. This helps us reformulate known results on singular locus of Schubert varieties.

Date published: Wednesday, March 8, 2023