On Tangent Cones of Schubert Varieties

被引：0
作者：
Fuchs D. [1 ]
Kirillov A. [2 ]
Morier-Genoud S. [3 ]
Ovsienko V. [4 ]
机构：
[1] University of California Davis, Mathematical Sciences Building One Shields Ave., Davis, 95616, CA
[2] Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, 19104-6395, PA
[3] Sorbonne Universités, UPMC Univ Paris 06, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ Paris Diderot, Sorbonne Paris Cité, Paris
[4] CNRS, Laboratoire de Mathématiques U.F.R. Sciences Exactes et Naturelles Moulin de la Housse, BP 1039, Reims Cedex 2
来源：
Arnold Mathematical Journal | 2017年 / 3卷 / 4期
关键词：
Essential set; Rank matrix; Schubert variety; Singularity; Tangent cone;
D O I：
10.1007/s40598-017-0074-x
中图分类号：
学科分类号：
摘要：
We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton’s essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix. © 2017, Institute for Mathematical Sciences (IMS), Stony Brook University, NY.
引用
收藏
页码：451 / 482
页数：31
相关论文
共 19 条
  • [1] Andre C.A.M., Basic sums of coadjoint orbits of the unitriangular group, J. Algebra, 176, pp. 959-1000, (1995)
  • [2] Billey S., Lakshmibai V., Singular Loci of Schubert Varieties, Progress in Mathematics, 182, (2000)
  • [3] Bochkarev M., Ignatyev M., Shevchenko A., Tangent cones to Schubert varieties in types A<sub>n</sub>, B<sub>n</sub> and C<sub>n</sub> , J. Algebra, 465, pp. 259-286, (2016)
  • [4] Brion M., Lectures on the Geometry of Flag Varieties, Topics in Cohomological Studies of Algebraic Varieties, pp, 33-85, (2005)
  • [5] Carrell J., Kuttler J., Singularities of Schubert varieties, tangent cones and Bruhat graphs, Am. J. Math., 128, 1, pp. 121-138, (2006)
  • [6] Eliseev D., Panov A., Tangent cones of Schubert varieties for A<sub>n</sub> of low rank, J. Math. Sci. (N. Y.), 188, 5, pp. 596-600, (2013)
  • [7] Eriksson K., Linusson S., Combinatorics of Fulton’s essential set, Duke Math. J., 85, pp. 61-76, (1996)
  • [8] Fulton W., Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., 65, pp. 381-420, (1992)
  • [9] Fulton W., Tableaux Y., With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts, vol. 35, (1997)
  • [10] Ignatyev M., Shevchenko A., On tangent cones to Schubert varieties in type D<sub>n</sub> , St. Petersbourg Math. J., 27, pp. 28-49, (2015)
12