A categorification of integral Specht modules
Authors:
Mikhail Khovanov, Volodymyr Mazorchuk and Catharina Stroppel
Journal:
Proc. Amer. Math. Soc. 136 (2008), 1163-1169
MSC (2000):
Primary 17B10, 05E10, 20C08
DOI:
https://doi.org/10.1090/S0002-9939-07-09124-1
Published electronically:
December 18, 2007
MathSciNet review:
2367090
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category for the Lie algebra
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- [Au] Maurice Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310. MR 349747, https://doi.org/10.1080/00927877408548230
- [BGS] 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, https://doi.org/10.1090/S0894-0347-96-00192-0
- [BG] J. N. Bernstein and S. I. Gel′fand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245–285. MR 581584
- [BFK] Joseph Bernstein, Igor Frenkel, and Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of 𝑈(𝔰𝔩₂) via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199–241. MR 1714141, https://doi.org/10.1007/s000290050047
- [BGG] I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of 𝔤-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
- [CR]
J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and
-categorification, to appear in Annals Math., math.RT/0407205.
- [Do] J. Matthew Douglass, An involution of the variety of flags fixed by a unipotent linear transformation, Adv. in Appl. Math. 17 (1996), no. 3, 357–379. MR 1406407, https://doi.org/10.1006/aama.1996.0015
- [FKS] Igor Frenkel, Mikhail Khovanov, and Catharina Stroppel, A categorification of finite-dimensional irreducible representations of quantum 𝔰𝔩₂ and their tensor products, Selecta Math. (N.S.) 12 (2006), no. 3-4, 379–431. MR 2305608, https://doi.org/10.1007/s00029-007-0031-y
- [Fu] 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
- [Ir] Ronald S. Irving, Projective modules in the category 𝒪_{𝒮}: self-duality, Trans. Amer. Math. Soc. 291 (1985), no. 2, 701–732. MR 800259, https://doi.org/10.1090/S0002-9947-1985-0800259-9
- [Ja] Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943
- [KL] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, https://doi.org/10.1007/BF01390031
- [MS] V. Mazorchuk and C. Stroppel, Projective-injective modules, Serre functors and symmetric algebras, to appear in J. reine angew. Math. math.RT/0508119.
- [Mu] G. E. Murphy, The representations of Hecke algebras of type 𝐴_{𝑛}, J. Algebra 173 (1995), no. 1, 97–121. MR 1327362, https://doi.org/10.1006/jabr.1995.1079
- [Na] Hiroshi Naruse, On an isomorphism between Specht module and left cell of 𝔖_{𝔫}, Tokyo J. Math. 12 (1989), no. 2, 247–267. MR 1030495, https://doi.org/10.3836/tjm/1270133181
- [R-C] Alvany Rocha-Caridi, Splitting criteria for 𝔤-modules induced from a parabolic and the Berňsteĭn-Gel′fand-Gel′fand resolution of a finite-dimensional, irreducible 𝔤-module, Trans. Amer. Math. Soc. 262 (1980), no. 2, 335–366. MR 586721, https://doi.org/10.1090/S0002-9947-1980-0586721-0
- [So] Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83–114. MR 1444322, https://doi.org/10.1090/S1088-4165-97-00021-6
- [St1] Catharina Stroppel, Category 𝒪: gradings and translation functors, J. Algebra 268 (2003), no. 1, 301–326. MR 2005290, https://doi.org/10.1016/S0021-8693(03)00308-9
- [St2] C. Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, math.RT/0608234.
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Additional Information
Mikhail Khovanov
Affiliation:
Department of Mathematics, Columbia University, New York, New York
Email:
khovanov@math.columbia.edu
Volodymyr Mazorchuk
Affiliation:
Department of Mathematics, Uppsala University, Uppsala, Sweden
Email:
mazor@math.uu.se
Catharina Stroppel
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow, United Kingdom
Email:
c.stroppel@maths.gla.ac.uk
DOI:
https://doi.org/10.1090/S0002-9939-07-09124-1
Received by editor(s):
September 14, 2006
Published electronically:
December 18, 2007
Additional Notes:
The first author was partially supported by the NSF grant DMS-0407784.
The second author was supported by STINT, the Royal Swedish Academy of Sciences, the Swedish Research Council and the MPI in Bonn.
The third author was supported by EPSRC grant 32199
Communicated by:
Dan Barbasch
Article copyright:
© Copyright 2007
American Mathematical Society