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Split subvarieties of group embeddings


Author: Nicolas Perrin
Journal: Trans. Amer. Math. Soc. 367 (2015), 8421-8438
MSC (2010): Primary 14M27; Secondary 20G15, 13A35
DOI: https://doi.org/10.1090/S0002-9947-2015-06279-5
Published electronically: March 20, 2015
MathSciNet review: 3403060
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Abstract: Let $ G$ be a connected reductive group and $ X$ an equivariant compactification of $ G$. In $ X$, we study generalised and opposite generalised Schubert varieties, and their intersections called generalised Richardson varieties and projected generalised Richardson varieties. Any complete $ G$-embedding has a canonical Frobenius splitting, and we prove that the compatibly split subvarieties are the generalised projected Richardson varieties extending a result of Knutson, Lam and Speyer to the situation.


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  • [1] Valery Alexeev and Michel Brion, Stable reductive varieties. I. Affine varieties, Invent. Math. 157 (2004), no. 2, 227-274. MR 2076923 (2005g:14088), https://doi.org/10.1007/s00222-003-0347-y
  • [2] Michel Brion, The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998), no. 1, 137-174. MR 1610599 (99b:14049), https://doi.org/10.1007/s000140050049
  • [3] Michel Brion, Multiplicity-free subvarieties of flag varieties, Commutative algebra (Grenoble/Lyon, 2001) Contemp. Math., vol. 331, Amer. Math. Soc., Providence, RI, 2003, pp. 13-23. MR 2011763 (2005c:14058), https://doi.org/10.1090/conm/331/05900
  • [4] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181 (33 #7330)
  • [5] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 0217086 (36 #178)
  • [6] Michel Brion and Shrawan Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2107324 (2005k:14104)
  • [7] Michel Brion and Patrick Polo, Large Schubert varieties, Represent. Theory 4 (2000), 97-126 (electronic). MR 1789463 (2001j:14066), https://doi.org/10.1090/S1088-4165-00-00069-8
  • [8] Michel Brion and Jesper Funch Thomsen, $ F$-regularity of large Schubert varieties, Amer. J. Math. 128 (2006), no. 4, 949-962. MR 2251590 (2007f:14047)
  • [9] William Fulton and Piotr Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998. Appendix J by the authors in collaboration with I. Ciocan-Fontanine. MR 1639468 (99m:14092)
  • [10] Xuhua He and Jesper Funch Thomsen, Geometry of $ B\times B$-orbit closures in equivariant embeddings, Adv. Math. 216 (2007), no. 2, 626-646. MR 2351372 (2008k:14092), https://doi.org/10.1016/j.aim.2007.06.001
  • [11] Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287-297. MR 0360616 (50 #13063)
  • [12] Allen Knutson, Thomas Lam, and David E. Speyer, Projections of Richardson varieties, J. Reine Angew. Math. 687 (2014), 133-157. MR 3176610, https://doi.org/10.1515/crelle-2012-0045
  • [13] Friedrich Knop, The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras, 1991, pp. 225-249. MR 1131314 (92m:14065)
  • [14] Friedrich Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv. 70 (1995), no. 2, 285-309. MR 1324631 (96c:14039), https://doi.org/10.1007/BF02566009
  • [15] V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27-40. MR 799251 (86k:14038), https://doi.org/10.2307/1971368
  • [16] Nicolas Perrin, On the geometry of spherical varieties, Transform. Groups 19 (2014), no. 1, 171-223. MR 3177371, https://doi.org/10.1007/s00031-014-9254-0
  • [17] Mohan S. Putcha, Linear algebraic monoids, London Mathematical Society Lecture Note Series, vol. 133, Cambridge University Press, Cambridge, 1988. MR 964690 (90a:20003)
  • [18] Mohan S. Putcha, Monoids on groups with $ BN$-pairs, J. Algebra 120 (1989), no. 1, 139-169. MR 977865 (89k:20091), https://doi.org/10.1016/0021-8693(89)90193-2
  • [19] Lex E. Renner, Linear algebraic monoids, Encyclopaedia of Mathematical Sciences, vol. 134, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, V. MR 2134980 (2006a:20002)
  • [20] K. Rietsch, Closure relations for totally nonnegative cells in $ G/P$, Math. Res. Lett. 13 (2006), no. 5-6, 775-786. MR 2280774 (2007j:14073), https://doi.org/10.4310/MRL.2006.v13.n5.a8
  • [21] A. Rittatore, Algebraic monoids and group embeddings, Transform. Groups 3 (1998), no. 4, 375-396. MR 1657536 (2000a:14056), https://doi.org/10.1007/BF01234534
  • [22] Alvaro Rittatore, Reductive embeddings are Cohen-Macaulay, Proc. Amer. Math. Soc. 131 (2003), no. 3, 675-684 (electronic). MR 1937404 (2004a:14048), https://doi.org/10.1090/S0002-9939-02-06843-0
  • [23] Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1-28. MR 0337963 (49 #2732)

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Additional Information

Nicolas Perrin
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
Email: perrin@math.uni-duesseldorf.de

DOI: https://doi.org/10.1090/S0002-9947-2015-06279-5
Received by editor(s): July 2, 2013
Received by editor(s) in revised form: September 15, 2013
Published electronically: March 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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