Abstract
We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato’s original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley–Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov’s arc algebra to the exotic setting.
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P. N. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (2008), no. 1, 27–62.
P. N. Achar, A. Henderson, E. Sommers, Pieces of nilpotent cones for classical groups, Represent. Theory 15 (2011), 584–616.
M. M. Asaeda, J. H. Przytycki, A. S. Sikora, Categorification of the Kauffman bracket skein module of I-bundles over surfaces, Algebr. Geom. Topol. 4 (2004), 1177–1210.
J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra I: cellularity, Mosc. Math. J. 11 (2011), no. 4, 685–722.
J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra III: category \( \mathcal{O} \), Represent. Theory 15 (2011), 170–243.
J. Brundan, C. Stroppel, Gradings on walled Brauer algebras and Khovanov’s arc algebra, Adv. Math. 231 (2012), no. 2, 709–773.
J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup, J. Eur. Math. Soc. 14 (2012), no. 2, 373–419.
H. Bao, W. Wang, H. Watanabe, Multiparameter quantum Schur duality of type B, Proc. Amer. Math. Soc. 146 (2018), no. 8, 3203–3216.
H. Can, Representations of the generalized symmetric groups, Beiträge Algebra Geom. 37 (1996), no. 2, 289–307.
N. Chriss, V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser Boston, Inc., Boston, MA, 1997.
S. Cautis, J. Kamnitzer, Knot homology via derived categories of coherent sheaves. I. The \( \mathfrak{sl} \)(2)-case, Duke Math. J. 142 (2008), 511–588.
Y. Chen, M. Khovanov, An invariant of tangle cobordisms via subquotients of arc rings, Fund. Math. 225 (2014), no. 1, 23–44.
D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.
M. Ehrig, C. Stroppel, 2-row Springer fibres and Khovanov diagram algebras for type D, Canadian J. Math. 68 (2016), no. 6, 1285–1333.
L. Fresse, A. Melnikov, On the singularity of the irreducible components of a Springer fiber in \( {\mathfrak{sl}}_n \), Selecta Math. 16 (2010), no. 3, 393–418.
F. 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.
P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994.
J. E. Grigsby, A. M. Licata, S. M. Wehrli, Annular Khovanov homology and knotted Schur–Weyl representations, Compos. Math. 154 (2018), no. 3, 459–502.
M. S. Im, C.-J. Lai, A. Wilbert, Irreducible components of two-row Springer fibers for all classical types, in preparation (2020).
S. Kato, An exotic Deligne–Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009), no. 2, 305–371.
S. Kato, Deformations of nilpotent cones and Springer correspondences, Amer. J. Math. 133 (2011), 519–553.
S. Kato, An algebraic study of extension algebras, Amer. J. Math. 139 (2017), no. 3, 567–615.
M. Khovanov, A functor-valued invariant of tangles, Alg. Geom. Top. 2 (2002), 665–741.
M. Khovanov, Crossingless matchings and the (n, n) Springer varieties, Commun. Contemporary Math. 6 (2004), 561–577.
D. Kazhdan, G. Lusztig, Proof of the Deligne–Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215.
A. Lacabanne, P. Vaz, Schur–Weyl duality, Verma modules, and row quotients of Ariki–Koike algebras, arXiv:2004.01065 (2020).
I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
P. Martin, H. Saleur, The blob algebra and the periodic Temperley–Lieb algebra, Lett. Math. Phys. 30 (1994), 189–206.
V. Mazorchuk, C. Stroppel, G(l, k, d)-modules via groupoids, J. Algebraic Combin. 43 (2016), no. 1, 11–32.
P. Martin, D. Woodcock, Generalized blob algebras and alcove geometry, LMS J. Comput. Math. 6 (2003), 249–296.
V. Nandakumar, Equivariant coherent sheaves on the exotic nilpotent cone, Represent. Theory 17 (2013), 663–681.
A. Nichols, V. Rittenberg, J. de Gier, One-boundary Temperley–Lieb algebras in the XXZ and loop models, J. Stat. Mech. Theor. Exp. (2005), P03003.
V. Nandakumar, D. Rosso, N. Saunders, Irreducible components of exotic Springer fibres II: the exotic Robinson–Schensted algorithm, arXiv:1710.08948 (2017).
V. Nandakumar, D. Rosso, N. Saunders, Irreducible components of exotic Springer fibres, J. Lond. Math. Soc. 98 (2018), no. 3, 609–637.
R. Orellana, A. Ram, Affine braids, Markov traces and the category \( \mathcal{O} \), in: Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math., Vol. 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 423–473.
D. Plaza, S. Ryom-Hansen, Graded cellular bases for Temperley–Lieb algebras of type A and B, J. Algebraic Combin. 14 (2014), no. 2, 137–177.
H. M. Russell, J. Tymoczko, Springer representations on the Khovanov Springer varieties, Math. Proc. Cambridge Philos. Soc. 151 (2011), 59–81.
D. Rose, D. Tubbenhauer, HOMFLYPT homology for links in handlebodies via type A Soergel bimodules, arXiv:1908.06878 (2019).
H. M. Russell, A topological construction for all two-row Springer varieties, Pacific J. Math. 253 (2011), 221–255.
G. Schäfer, A graphical calculus for 2-block Spaltenstein varieties, Glasgow Math. J. 54 (2012), 449–477.
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42. Springer-Verlag, New York, 1977.
T. Shoji, On the Springer representations of the Weyl groups of classical algebraic groups, Comm. in Algebra 7 (1979), 1713–1745.
N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold, Nederl. Akad. Wetensch. Proc. Ser. A 79 (1976), 452–456.
N. Spaltenstein, Classes Unipotentes et Sous-groupes de Borel, Lecture Notes in Mathematics, Vol. 946, Springer-Verlag, Berlin, 1982.
T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207.
T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), 279–293.
T. Shoji, K. Sorlin, Exotic symmetric space over a finite field, II, Transform. Groups 19 (2014), no. 3, 887–926.
C. Stroppel, Parabolic category \( \mathcal{O} \), perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math. 145 (2009), no. 4, 954–992.
C. Stroppel, B. Webster, 2-block Springer fibers: convolution algebras and coherent sheaves, Comment. Math. Helv. 87 (2012), 477–520.
C. Stroppel, A. Wilbert, Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory, Math. Z. 292 (2019), no. 3–4, 1387–1430.
T. tom Dieck, Symmetrische Brücken und Knotentheorie zu den Dynkin-Diagrammen vom Typ B, J. Reine Angew. Math 451 (1994), 71–88.
J. A. Vargas, Fixed points under the action of unipotent elements of SLn in the flag variety, Bol. Soc. Mat. Mexicana 24 (1979), no. 1, 1–14.
M. van Leeuwen, A Robinson–Schensted Algorithm in the Geometry of Flags for Classical Groups, PhD thesis, Rijksuniversiteit Utrecht, 1989.
S. Wehrli, A remark on the topology of (n,n) Springer varieties, arXiv:0908.2185 (2009).
A. Wilbert, Topology of two-row Springer fibers for the even orthogonal and symplectic group, Trans. Amer. Math. Soc. 370 (2018), 2707–2737.
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SAUNDERS, N., WILBERT, A. EXOTIC SPRINGER FIBERS FOR ORBITS CORRESPONDING TO ONE-ROW BIPARTITIONS. Transformation Groups 27, 1111–1147 (2022). https://doi.org/10.1007/s00031-020-09613-0
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DOI: https://doi.org/10.1007/s00031-020-09613-0