The multiplicity of a singularity in a vexillary Schubert variety

TL;DR

This paper derives a combinatorial formula for the Hilbert-Samuel multiplicity at points on vexillary Schubert varieties in classical flag varieties, extending previous results and providing new proofs.

Contribution

It introduces a new combinatorial formula using excited Young diagrams for multiplicities on vexillary Schubert varieties, generalizing prior work and offering a novel proof of a key theorem.

Findings

Formula expressed via excited Young diagrams

Extension of known results to broader classes of Schubert varieties

New proof of Li-Yong's theorem in type A vexillary case

Abstract

In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is expressed in terms of excited Young diagrams, and extends results for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for the maximal isotropic (symplectic and orthogonal) Grassmannians to Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also provide a new proof of a theorem of Li-Yong in the type A vexillary case. The main ingredient is an isomorphism between certain neighborhoods of fixed points, known as Kazhdan-Lusztig varieties, which, in turn, relies on a direct sum embedding previously used by Anderson-Fulton to relate vexillary loci to Grassmannian loci.