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- Title
Generic singularities of certain Schubert varieties.
- Authors
Brion, Michel; Polo, Patrick
- Abstract
Let G be a connected semisimple algebraic group, B a Borel subgroup, T a maximal torus in B with Weyl group W, and Q a subgroup containing B. For $w \in W$ , let $X_{wQ}$ denote the Schubert variety $\overline{BwQ}/Q$ . For $y\in W$ such that $X_{yQ}\subseteq X_{wQ}$ , one knows that ByQ / Q admits a T-stable transversal in $X_{wQ}$ , which we denote by ${\cal N}_{yQ,wQ}$ . We prove that, under certain hypotheses, ${\cal N}_{yQ, wQ}$ is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and $X_{yQ}$ is an irreducible component of the boundary of $X_{wQ}$ (that is, the complement of the open orbit of the stabiliser in G of $X_{wQ}$ ). As a consequence, we describe the singularity of $X_{wQ}$ along ByQ / Q and obtain that the boundary of $X_{wQ}$ equals its singular locus.
- Publication
Mathematische Zeitschrift, 1999, Vol 231, Issue 2, p301
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/PL00004729