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Transactions of the American Mathematical Society

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Topology of two-row Springer fibers for the even orthogonal and symplectic group


Author: Arik Wilbert
Journal: Trans. Amer. Math. Soc. 370 (2018), 2707-2737
MSC (2010): Primary 14M15; Secondary 05E10, 17B08, 20C08
DOI: https://doi.org/10.1090/tran/7194
Published electronically: September 15, 2017
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Abstract: We define an explicit topological model for every two-row Springer fiber associated with the even orthogonal group and prove that the respective topological model is homeomorphic to its corresponding Springer fiber. This confirms a conjecture by Ehrig and Stroppel concerning the topology of the equal-row Springer fiber for the even orthogonal group. Moreover, we show that every two-row Springer fiber for the symplectic group is homeomorphic (even isomorphic as an algebraic variety) to a connected component of a certain two-row Springer fiber for the even orthogonal group.


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  • [1] Sabin Cautis and Joel Kamnitzer, Knot homology via derived categories of coherent sheaves. I. The $ {\mathfrak{sl}}(2)$-case, Duke Math. J. 142 (2008), no. 3, 511-588. MR 2411561, https://doi.org/10.1215/00127094-2008-012
  • [2] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
  • [3] Michael Ehrig and Catharina Stroppel, 2-row Springer fibres and Khovanov diagram algebras for type D, Canad. J. Math. 68 (2016), no. 6, 1285-1333. MR 3563723, https://doi.org/10.4153/CJM-2015-051-4
  • [4] Michael Ehrig and Catharina Stroppel, Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians, Selecta Math. (N.S.) 22 (2016), no. 3, 1455-1536. MR 3518556, https://doi.org/10.1007/s00029-015-0215-9
  • [5] 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, https://doi.org/10.1016/S0001-8708(02)00072-5
  • [6] Murray Gerstenhaber, Dominance over the classical groups, Ann. of Math. (2) 74 (1961), 532-569. MR 0136683, https://doi.org/10.2307/1970297
  • [7] Anthony Henderson and Anthony Licata, Diagram automorphisms of quiver varieties, Adv. Math. 267 (2014), 225-276. MR 3269179, https://doi.org/10.1016/j.aim.2014.08.007
  • [8] Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. MR 1740682, https://doi.org/10.1215/S0012-7094-00-10131-7
  • [9] Mikhail Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665-741. MR 1928174, https://doi.org/10.2140/agt.2002.2.665
  • [10] Mikhail Khovanov, Crossingless matchings and the cohomology of $ (n,n)$ Springer varieties, Commun. Contemp. Math. 6 (2004), no. 4, 561-577. MR 2078414, https://doi.org/10.1142/S0219199704001471
  • [11] Tobias Lejczyk and Catharina Stroppel, A graphical description of $ (D_n,A_{n-1})$ Kazhdan-Lusztig polynomials, Glasg. Math. J. 55 (2013), no. 2, 313-340. MR 3040865, https://doi.org/10.1017/S0017089512000547
  • [12] Thomas Pietraho, Components of the Springer fiber and domino tableaux, J. Algebra 272 (2004), no. 2, 711-729. MR 2028078, https://doi.org/10.1016/S0021-8693(03)00434-4
  • [13] Heather M. Russell, A topological construction for all two-row Springer varieties, Pacific J. Math. 253 (2011), no. 1, 221-255. MR 2869443, https://doi.org/10.2140/pjm.2011.253.221
  • [14] Heather M. Russell and Julianna S. Tymoczko, Springer representations on the Khovanov Springer varieties, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 1, 59-81. MR 2801314, https://doi.org/10.1017/S0305004111000132
  • [15] P. Slodowy, Platonic solids, Kleinian singularities, and Lie groups, Algebraic geometry (Ann Arbor, Mich., 1981) Lecture Notes in Math., vol. 1008, Springer, Berlin, 1983, pp. 102-138. MR 723712, https://doi.org/10.1007/BFb0065703
  • [16] N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 5, 452-456. MR 0485901
  • [17] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610
  • [18] Catharina Stroppel and Ben Webster, 2-block Springer fibers: convolution algebras and coherent sheaves, Comment. Math. Helv. 87 (2012), no. 2, 477-520. MR 2914857, https://doi.org/10.4171/CMH/261
  • [19] M. van Leeuwen, A Robinson-Schensted algorithm in the geometry of flags for classical groups, PhD thesis, Rijksuniversiteit Utrecht, 1989.
  • [20] J. A. Vargas, Fixed points under the action of unipotent elements of $ {\rm SL}_{n}$ in the flag variety, Bol. Soc. Mat. Mexicana (2) 24 (1979), no. 1, 1-14. MR 579665
  • [21] S. Wehrli, A remark on the topology of (n,n) Springer varieties, arXiv:0908.2185, 2009.
  • [22] John Williamson, The Conjunctive Equivalence of Pencils of Hermitian and Anti-Hermitian Matrices, Amer. J. Math. 59 (1937), no. 2, 399-413. MR 1507253, https://doi.org/10.2307/2371425

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Additional Information

Arik Wilbert
Affiliation: Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: wilbert@math.uni-bonn.de

DOI: https://doi.org/10.1090/tran/7194
Received by editor(s): March 13, 2016
Received by editor(s) in revised form: October 17, 2016
Published electronically: September 15, 2017
Additional Notes: This research was funded by a Hausdorff scholarship of the Bonn International Graduate School in Mathematics
Article copyright: © Copyright 2017 American Mathematical Society

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