Tangent cones to Schubert varieties in types An, Bn and Cn

Journal of Algebra. 2016. Vol. 465. No. November . P. 259–286.

Bochkarev M., Ignatyev M. V., Shevchenko A. A.

We study tangent cones to Schubert subvarieties of the flag variety of a complex reductive group G. Let T be a maximal torus of G, B be a Borel subgroup of G containing T, Φ be the root system of G with respect to T, W be the Weyl group of Φ, and F=G/B be the flag variety. We prove that if every irreducible component of Φ is of type Bn or Cn, and w1, w2 are two distinct involutions in W, then the tangent cones at the point p=eB to the corresponding Schubert subvarieties Xw1, Xw2 of F do not coincide as subschemes of the tangent space TpF. We also show that if every irreducible component of Φ is of type An or Cn, then the reduced tangent cones to Xw1 and Xw2 do not coincide as subvarieties of TpF. 

Keywords: Flag variety, Involution in the Weyl group, Kostant–Kumar polynomial, Reduced tangent cone, Schubert variety, Tangent cone