An equivariant basis for the cohomology of Springer fibers
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- by Martha Precup and Edward Richmond;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 481-509
- DOI: https://doi.org/10.1090/btran/57
- Published electronically: June 10, 2021
- HTML | PDF
Abstract:
Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for $GL_n(\mathbb {C})$ using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.References
- Hiraku Abe and Tatsuya Horiguchi, The torus equivariant cohomology rings of Springer varieties, Topology Appl. 208 (2016), 143–159. MR 3506975, DOI 10.1016/j.topol.2016.05.004
- Sara Billey and Mark Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), no. 2, 443–482. MR 1290232, DOI 10.1090/S0894-0347-1995-1290232-1
- Sara C. Billey, Kostant polynomials and the cohomology ring for $G/B$, Duke Math. J. 96 (1999), no. 1, 205–224. MR 1663931, DOI 10.1215/S0012-7094-99-09606-0
- Sara C. Billey, William Jockusch, and Richard P. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), no. 4, 345–374. MR 1241505, DOI 10.1023/A:1022419800503
- Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
- James B. Carrell, Orbits of the Weyl group and a theorem of DeConcini and Procesi, Compositio Math. 60 (1986), no. 1, 45–52. MR 867955
- C. De Concini, G. Lusztig, and C. Procesi, Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), no. 1, 15–34. MR 924700, DOI 10.1090/S0894-0347-1988-0924700-2
- Corrado De Concini and Claudio Procesi, Symmetric functions, conjugacy classes and the flag variety, Invent. Math. 64 (1981), no. 2, 203–219. MR 629470, DOI 10.1007/BF01389168
- Barry Dewitt and Megumi Harada, Poset pinball, highest forms, and $(n-2,2)$ Springer varieties, Electron. J. Combin. 19 (2012), no. 1, Paper 56, 35. MR 2900431, DOI 10.37236/2126
- Elizabeth Drellich, Monk’s rule and Giambelli’s formula for Peterson varieties of all Lie types, J. Algebraic Combin. 41 (2015), no. 2, 539–575. MR 3306081, DOI 10.1007/s10801-014-0545-2
- David S. Dummit and Richard M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR 2286236
- David Eisenbud and Joe Harris, 3264 and all that—a second course in algebraic geometry, Cambridge University Press, Cambridge, 2016. MR 3617981, DOI 10.1017/CBO9781139062046
- Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793, DOI 10.1006/aima.1994.1009
- Lucas Fresse, Betti numbers of Springer fibers in type $A$, J. Algebra 322 (2009), no. 7, 2566–2579. MR 2553695, DOI 10.1016/j.jalgebra.2009.07.008
- Lucas Fresse, Singular components of Springer fibers in the two-column case, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2429–2444 (English, with English and French summaries). MR 2640925, DOI 10.5802/aif.2495
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
- Francis Y. C. Fung, On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory, Adv. Math. 178 (2003), no. 2, 244–276. MR 1994220, DOI 10.1016/S0001-8708(02)00072-5
- A. M. Garsia and C. Procesi, On certain graded $S_n$-modules and the $q$-Kostka polynomials, Adv. Math. 94 (1992), no. 1, 82–138. MR 1168926, DOI 10.1016/0001-8708(92)90034-I
- William Graham and R. Zierau, Smooth components of Springer fibers, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 5, 2139–2182 (2012) (English, with English and French summaries). MR 2961851, DOI 10.5802/aif.2669
- Megumi Harada and Julianna Tymoczko, A positive Monk formula in the $S^1$-equivariant cohomology of type $A$ Peterson varieties, Proc. Lond. Math. Soc. (3) 103 (2011), no. 1, 40–72. MR 2812501, DOI 10.1112/plms/pdq038
- Megumi Harada and Julianna Tymoczko, Poset pinball, GKM-compatible subspaces, and Hessenberg varieties, J. Math. Soc. Japan 69 (2017), no. 3, 945–994. MR 3685032, DOI 10.2969/jmsj/06930945
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 79 (1964), 205–326. MR 199184, DOI 10.2307/1970547
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
- Hanspeter Kraft, Conjugacy classes and Weyl group representations, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980) Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 191–205. MR 646820
- Shrawan Kumar and Claudio Procesi, An algebro-geometric realization of equivariant cohomology of some Springer fibers, J. Algebra 368 (2012), 70–74. MR 2955222, DOI 10.1016/j.jalgebra.2012.06.019
- Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739
- I. G. Macdonald, Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991) London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 73–99. MR 1161461
- Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow; Cours Spécialisés [Specialized Courses], 3. MR 1852463
- Aba Mbirika, A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties, Electron. J. Combin. 17 (2010), no. 1, Research Paper 153, 29. MR 2745706, DOI 10.37236/425
- Aba Mbirika and Julianna Tymoczko, Generalizing Tanisaki’s ideal via ideals of truncated symmetric functions, J. Algebraic Combin. 37 (2013), no. 1, 167–199. MR 3016306, DOI 10.1007/s10801-012-0372-2
- Martha Precup and Julianna Tymoczko, Springer fibers and Schubert points, European J. Combin. 76 (2019), 10–26. MR 3886508, DOI 10.1016/j.ejc.2018.08.010
- Naohisa Shimomura, A theorem on the fixed point set of a unipotent transformation on the flag manifold, J. Math. Soc. Japan 32 (1980), no. 1, 55–64. MR 554515, DOI 10.2969/jmsj/03210055
- N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold, Indag. Math. 38 (1976), no. 5, 452–456. Nederl. Akad. Wetensch. Proc. Ser. A 79. MR 485901, DOI 10.1016/S1385-7258(76)80008-X
- T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207. MR 442103, DOI 10.1007/BF01390009
- T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293. MR 491988, DOI 10.1007/BF01403165
- Toshiyuki Tanisaki, Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups, Tohoku Math. J. (2) 34 (1982), no. 4, 575–585. MR 685425, DOI 10.2748/tmj/1178229158
- Julianna S. Tymoczko, Linear conditions imposed on flag varieties, Amer. J. Math. 128 (2006), no. 6, 1587–1604. MR 2275912, DOI 10.1353/ajm.2006.0050
Bibliographic Information
- Martha Precup
- Affiliation: Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, Missouri 63130
- MR Author ID: 1043988
- Email: martha.precup@wustl.edu
- Edward Richmond
- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma , 74078
- MR Author ID: 875224
- Email: edward.richmond@okstate.edu
- Received by editor(s): February 4, 2020
- Received by editor(s) in revised form: August 14, 2020
- Published electronically: June 10, 2021
- Additional Notes: The first author was supported by an Oklahoma State University CAS summer research grant. The second author was partially supported by an AWM-NSF travel grant and NSF grant DMS 1954001 during the course of this research.
- © Copyright 2021 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 481-509
- MSC (2020): Primary 05E10, 14M15; Secondary 14N15
- DOI: https://doi.org/10.1090/btran/57
- MathSciNet review: 4273195