On tangent cones to Schubert varieties in type $E$

Ignatyev, Mikhail V. ; Shevchenko, Aleksandr A.

Communications in Mathematics, Tome 28 (2020) no. 2, pp. 179-197

Abstract (VO)

We consider tangent cones to Schubert subvarieties of the flag variety $G/B$, where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_6$, $E_7$ or $E_8$. We prove that if $w_1$ and $w_2$ form a good pair of involutions in the Weyl group $W$ of $G$ then the tangent cones $C_{w_1}$ and $C_{w_2}$ to the corresponding Schubert subvarieties of $G/B$ do not coincide as subschemes of the tangent space to $G/B$ at the neutral point.

Détail

Classification : 14M15, 17B22
Keywords: flag variety; Schubert variety; tangent cone; involution in the Weyl group; Kostant-Kumar polynomial

Bibliographie

[1] Billey, S.C.: Kostant polynomials and the cohomology ring for $G/B$. Duke Mathematical Journal, 96, 1, 1999, 205-224, Citeseer, | DOI | MR

[2] Bjorner, A., Brenti, F.: Combinatorics of Coxeter groups. 231, 2005, Springer Science & Business Media, Graduate Texts in Mathematics {231}. | MR

[3] Bochkarev, M.A., Ignatyev, M.V., Shevchenko, A.A.: Tangent cones to Schubert varieties in types $A_n$, $B_n$ and $C_n$. Journal of Algebra, 465, 2016, 259-286, Elsevier, | DOI | MR

[4] Bourbaki, N.: Lie groups and Lie algebras. Chapters 4--6. 2002, Springer-Verlag, Berlin, Translated from the 1968 French original. | MR

[5] Sarason, I.G., Billey, S., Sarason, S., Lakshmibai, V.: Singular loci of Schubert varieties. 182, 2000, Springer Science & Business Media, | MR

[6] Deodhar, V.V.: On the root system of a Coxeter group. Communications in Algebra, 10, 6, 1982, 611-630, Taylor & Francis, | DOI | MR

[7] Dyer, M.J.: The nil Hecke ring and Deodhar's conjecture on Bruhat intervals. Inventiones Mathematicae, 111, 1, 1993, 571-574, Springer, | DOI | MR

[8] Eliseev, D.Yu., Panov, A.N.: Tangent cones of Schubert varieties for $A_n$ of lower rank (in Russian). Zapiski Nauchnykh Seminarov POMI, 394, 2011, 218-225, St. Petersburg Department of Steklov Institute of Mathematics, Russian, English transl.: Journal of Mathematical Sciences 188 (5) (2013), 596--600.. | MR

[9] Eliseev, D.Yu., Ignatyev, M.V.: Kostant-Kumar polynomials and tangent cones to Schubert varieties for involutions in $A_n$, $F_4$ and $G_2$ (in Russian). Zapiski Nauchnykh Seminarov POMI, 414, 2013, 82-105, Springer, English transl.: Journal of Mathematical Sciences {199} (3) (2014), 289--301.. | MR

[10] Humphreys, J.E.: Linear algebraic groups. 1975, Springer, | MR

[11] Humphreys, J.E.: Reflection groups and Coxeter groups. 1992, Cambridge University Press, | MR

[12] Ignatyev, M.V., Shevchenko, A.A.: On tangent cones to Schubert varieties in type $D_n$ (in Russian). Algebra i Analiz, 27, 4, 2015, 28-49, English transl.: St. Petersburg Mathematical Journal 27 (4) (2016), 609--623.. | MR

[13] Kostant, B., Kumar, S.: The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$. Proceedings of the National Academy of Sciences, 83, 6, 1986, 1543-1545, National Acad Sciences, | MR

[14] Kostant, B., Kumar, S.: $T$-equivariant $K$-theory of generalized flag varieties. Journal of Differential Geometry, 32, 2, 1990, 549-603, Lehigh University, | MR

[15] Kumar, S.: The nil Hecke ring and singularity of Schubert varieties. Inventiones Mathematicae, 123, 3, 1996, 471-506, Springer, | DOI | MR

[16] Springer, T.A.: Some remarks on involutions in Coxeter groups. Communications in Algebra, 10, 6, 1982, 631-636, Taylor & Francis, | DOI | MR

[17] al., W.A. Stein et: Sage Mathematics Software (Version 9.1). 2020, The Sage {Development} Team, Available at { http://www.sagemath.org}