Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

A diagram algebra for Soergel modules corresponding to smooth Schubert varieties


Author: Antonio Sartori
Journal: Trans. Amer. Math. Soc. 368 (2016), 889-938
MSC (2010): Primary 16W50; Secondary 13F20, 05E05, 17B10
DOI: https://doi.org/10.1090/tran/6346
Published electronically: May 11, 2015
MathSciNet review: 3430353
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module categories are equivalent to the subquotient categories of the BGG category $ \mathcal {O}(\mathfrak{gl}_n)$ which show up in categorification of $ \mathfrak{gl}(1\vert 1)$-representations. We construct diagrammatically the graded cellular structure and the properly stratified structure of these algebras.


References [Enhancements On Off] (What's this?)

  • [AM11] Troels Agerholm and Volodymyr Mazorchuk, On selfadjoint functors satisfying polynomial relations, J. Algebra 330 (2011), 448-467. MR 2774639 (2012e:16041), https://doi.org/10.1016/j.jalgebra.2011.01.004
  • [Bac01] Erik Backelin, The Hom-spaces between projective functors, Represent. Theory 5 (2001), 267-283 (electronic). MR 1857082 (2002f:17007), https://doi.org/10.1090/S1088-4165-01-00099-1
  • [BG80] J. N. Bernstein and S. I. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245-285. MR 581584 (82c:17003)
  • [BGG76] I. N. Bernšteĭn, I. M. Gelfand, and S. I. Gelfand, A certain category of $ {\germ g}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1-8 (Russian). MR 0407097 (53 #10880)
  • [BGS96] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010), https://doi.org/10.1090/S0894-0347-96-00192-0
  • [BJS93] 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 (94m:05197), https://doi.org/10.1023/A:1022419800503
  • [Bru08] Jonathan Brundan, Symmetric functions, parabolic category $ \mathcal {O}$, and the Springer fiber, Duke Math. J. 143 (2008), no. 1, 41-79. MR 2414744 (2009h:17007), https://doi.org/10.1215/00127094-2008-015
  • [BS10] Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov's diagram algebra. II. Koszulity, Transform. Groups 15 (2010), no. 1, 1-45. MR 2600694 (2011b:17014), https://doi.org/10.1007/s00031-010-9079-4
  • [BS11a] Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov's diagram algebra I: cellularity, Mosc. Math. J. 11 (2011), no. 4, 685-722, 821-822 (English, with English and Russian summaries). MR 2918294
  • [BS11b] Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov's diagram algebra III: category $ \mathcal {O}$, Represent. Theory 15 (2011), 170-243. MR 2781018 (2012b:17016), https://doi.org/10.1090/S1088-4165-2011-00389-7
  • [BS12a] Jonathan Brundan and Catharina Stroppel, Gradings on walled Brauer algebras and Khovanov's arc algebra, Adv. Math. 231 (2012), no. 2, 709-773. MR 2955190, https://doi.org/10.1016/j.aim.2012.05.016
  • [BS12b] Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 373-419. MR 2881300 (2012m:17009), https://doi.org/10.4171/JEMS/306
  • [CLO07] David Cox, John Little, and Donal O'Shea, Ideals, varieties, and algorithms, An introduction to computational algebraic geometry and commutative algebra, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. MR 2290010 (2007h:13036)
  • [Dem73] Michel Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301 (French). MR 0342522 (49 #7268)
  • [DM07] Yuriy Drozd and Volodymyr Mazorchuk, Koszul duality for extension algebras of standard modules, J. Pure Appl. Algebra 211 (2007), no. 2, 484-496. MR 2341265 (2008j:16083), https://doi.org/10.1016/j.jpaa.2007.01.014
  • [Du05] Jie Du, Robinson-Schensted algorithm and Vogan equivalence, J. Combin. Theory Ser. A 112 (2005), no. 1, 165-172. MR 2167481 (2006m:05253), https://doi.org/10.1016/j.jcta.2005.01.008
  • [EK09] Ben Elias and Mikhail Khovanov, Diagrammatics for Soergel categories, Int. J. Math. Math. Sci. (2010), Art. ID 978635, 58. MR 3095655
  • [EW12] Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089-1136. MR 3245013, https://doi.org/10.4007/annals.2014.180.3.6
  • [FK97] Igor B. Frenkel and Mikhail G. Khovanov, Canonical bases in tensor products and graphical calculus for $ U_q({\mathfrak{s}}{\mathfrak{l}}_2)$, Duke Math. J. 87 (1997), no. 3, 409-480. MR 1446615 (99a:17019), https://doi.org/10.1215/S0012-7094-97-08715-9
  • [FKM02] V. Futorny, S. König, and V. Mazorchuk, Categories of induced modules and standardly stratified algebras, Algebr. Represent. Theory 5 (2002), no. 3, 259-276. MR 1921761 (2003g:17005), https://doi.org/10.1023/A:1016579318115
  • [Fri07] Anders Frisk, Dlab's theorem and tilting modules for stratified algebras, J. Algebra 314 (2007), no. 2, 507-537. MR 2344576 (2009b:16028), https://doi.org/10.1016/j.jalgebra.2006.08.041
  • [Ful97] William Fulton, Young tableaux, With applications to representation theory and geometry, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. MR 1464693 (99f:05119)
  • [GL96] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1-34. MR 1376244 (97h:20016), https://doi.org/10.1007/BF01232365
  • [GR02] V. Gasharov and V. Reiner, Cohomology of smooth Schubert varieties in partial flag manifolds, J. London Math. Soc. (2) 66 (2002), no. 3, 550-562. MR 1934291 (2003i:14064), https://doi.org/10.1112/S0024610702003605
  • [HM10] Jun Hu and Andrew Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type $ A$, Adv. Math. 225 (2010), no. 2, 598-642. MR 2671176 (2011g:20006), https://doi.org/10.1016/j.aim.2010.03.002
  • [Hum08] James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $ \mathcal {O}$, Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR 2428237 (2009f:17013)
  • [Kho00] Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. MR 1740682 (2002j:57025), https://doi.org/10.1215/S0012-7094-00-10131-7
  • [Kho04] Mikhail Khovanov, Crossingless matchings and the cohomology of $ (n,n)$ Springer varieties, Commun. Contemp. Math. 6 (2004), no. 4, 561-577. MR 2078414 (2005g:14090), https://doi.org/10.1142/S0219199704001471
  • [KL79] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 560412 (81j:20066), https://doi.org/10.1007/BF01390031
  • [KL80] David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185-203. MR 573434 (84g:14054)
  • [Knu73] Donald E. Knuth, The art of computer programming. Volume 3, Sorting and searching; Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. MR 0445948 (56 #4281)
  • [Mac91] 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 (93d:05159)
  • [Maz04] Volodymyr Mazorchuk, Stratified algebras arising in Lie theory, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun., vol. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 245-260. MR 2057398 (2005b:17013)
  • [MS08] Volodymyr Mazorchuk and Catharina Stroppel, Projective-injective modules, Serre functors and symmetric algebras, J. Reine Angew. Math. 616 (2008), 131-165. MR 2369489 (2009e:16027), https://doi.org/10.1515/CRELLE.2008.020
  • [Sar13] A. Sartori, Categorification of tensor powers of the vector representation of $ U_q(\mathfrak{gl}(1\vert 1))$, arXiv e-prints (2013), 1305.6162.
  • [Sar14] A. Sartori, Ph.D. thesis, Universität Bonn, 2014.
  • [Soe90] Wolfgang Soergel, Kategorie $ \mathcal {O}$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421-445 (German, with English summary). MR 1029692 (91e:17007), https://doi.org/10.2307/1990960
  • [Soe92] Wolfgang Soergel, The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992), 49-74. MR 1173115 (94b:17011), https://doi.org/10.1515/crll.1992.429.49
  • [Soe97] Wolfgang Soergel, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Represent. Theory 1 (1997), 37-68 (electronic) (German, with English summary). MR 1445511 (99d:17023), https://doi.org/10.1090/S1088-4165-97-00006-X
  • [Str03] Catharina Stroppel, Category $ \mathcal {O}$: quivers and endomorphism rings of projectives, Represent. Theory 7 (2003), 322-345 (electronic). MR 2017061 (2004h:17007), https://doi.org/10.1090/S1088-4165-03-00152-3
  • [Str09] Catharina Stroppel, Parabolic category $ \mathcal {O}$, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math. 145 (2009), no. 4, 954-992. MR 2521250 (2011a:17014), https://doi.org/10.1112/S0010437X09004035
  • [SW12] 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
  • [Tan82] Toshiyuki Tanisaki, Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups, Tôhoku Math. J. (2) 34 (1982), no. 4, 575-585. MR 685425 (84g:14049), https://doi.org/10.2748/tmj/1178229158
  • [Wil11] Geordie Williamson, Singular Soergel bimodules, Int. Math. Res. Not. IMRN 20 (2011), 4555-4632. MR 2844932, https://doi.org/10.1093/imrn/rnq263

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 16W50, 13F20, 05E05, 17B10

Retrieve articles in all journals with MSC (2010): 16W50, 13F20, 05E05, 17B10


Additional Information

Antonio Sartori
Affiliation: Mathematisches Institut, Endenicher Allee 60, Universität Bonn, 53115 Bonn, Germany
Address at time of publication: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany
Email: antonio.sartori@math.uni-freiburg.de

DOI: https://doi.org/10.1090/tran/6346
Keywords: Diagram algebra, symmetric polynomials, category $\mathcal{O}$, Soergel modules, Khovanov algebra
Received by editor(s): October 14, 2013
Received by editor(s) in revised form: December 4, 2013
Published electronically: May 11, 2015
Additional Notes: This work has been supported by the Graduiertenkolleg 1150, funded by the Deutsche Forschungsgemeinschaft.
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society