The multiplicity of a singularity in a vexillary Schubert variety
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.
Keywords
Cite
@article{arxiv.2112.07375,
title = {The multiplicity of a singularity in a vexillary Schubert variety},
author = {David Anderson and Takeshi Ikeda and Minyoung Jeon and Ryotaro Kawago},
journal= {arXiv preprint arXiv:2112.07375},
year = {2021}
}