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arXiv · 2112.07375

The multiplicity of a singularity in a vexillary Schubert variety

David Anderson·Takeshi Ikeda·Minyoung Jeon·Ryotaro Kawago
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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.

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David Anderson, Takeshi Ikeda, Minyoung Jeon, Ryotaro Kawago. 2021-12-14. The multiplicity of a singularity in a vexillary Schubert variety. https://arxiv.org/abs/2112.07375

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