Kostant--Kumar polynomials and tangent cones to Schubert varieties for involutions in A_n, F₄ and G₂

2012/10/21 by Mikhail V. Ignatyev, Dmitriy Y. Eliseev · 1 citation
Mathematics · #math.RT #math.AG #msc:16G30 #msc:20G20

paper · pdf

published as J. Math. Sci. 199 (2014), no. 3, 289-301 · 15 pages

arxiv created 2012/10/21 · arxiv updated 2014/10/16

Abstract

Let G be a reductive complex algebraic group, T a maximal torus of G, B a Borel subgroup of G containing T, Φ the root system of G w.r.t. T, W the Weyl group of Φ. Denote by \Fo = G/B the flag variety, by X_w the Schubert subvariety of \Fo associated with an element w∈ W, and by C_w the tangent cone to X_w at the point p = eB. Then C_w is a subscheme of the tangent space T_pX_w⊆ T_p\Fo. Suppose w, w' are distinct involutions in W. Using the so-called Kostant--Kumar polynomials, we show that if every irreducible component of Φ is of type A_n, F₄ or G₂, then C_w and C_w' do not coincide.

Cited by

• Tangent cones to Schubert varieties in types A_n, B_n and C_n