Generic singularities of certain Schubert varieties

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 ∈ W , let X wQ denote the Schubert variety BwQ/Q. For y ∈ W such that X yQ ⊆ X wQ , one knows that ByQ/Q admits a T -stable transversal in X wQ , which we denote by N yQ,wQ . We prove that, under certain hypotheses, 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.

Hilbert functions of points on Schubert varieties in the symplectic Grassmannian

“…By Proposition 5.10, the multiplicity of the diagonal element in B j is even (possibly 0). so that S w = {(9, 3), (10,2), (17,13), (18,12), (21,20), (25,22), (27,26), (28,19), (34, 30), (35, 29), (36, 11), (44, 38), (45, 37), (46, 1)}.…”

“…Recursive formulas for the multiplicity and for the Hilbert function in the case of minuscule G/P and also in the case of symplectic Grassmannian were obtained by Lakshmibai and Weyman [22] in 1990. The singularities of Schubert varieties in the symplectic Grassmannian have also been studied by Brion and Polo [2] who determine the multiplicity at varieties has proceeded almost in parallel to the study of Schubert varieties, but often as an independent pursuit. In particular, explicit formulas for the multiplicity and the Hilbert function for various classes of determinantal varieties have been obtained.…”

Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

“…Assume first that the C is an outer circle, i.e., it is not contained in any other circle, and there are no dotted circles to the right of C and no lines to the left of C. 5 Then, there exists a unique μ ∈ such that λ → μ and such that the end points of γ are precisely the positions where λ and μ differ, and this is given by a local move as displayed in the last row of (3.14). Checking the multiplication, one sees that in this case,…”

Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire

“…[3], 3.3 et 4.6, voir aussi [4]) suggèrent de définir les deux types de singularité suivants. Soit X v une composante irréductible du lieu singulier de X w ; on dira qu'elle est de type S 1 s'il existe des entiers s et t (s, t ≥ 2) tels que N v,w soit isomorpheà la variété C s,t des matrices de taille (s, t) et de rang au plus 1, et l'on dira que X v est de type S 2 si N v,w est isomorpheà un cône quadratique non dégénéré de dimension au moins 5.…”