Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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 $ \mathcal{O}$ for the Lie algebra $ \mathfrak{sl}_n(\mathbb{C})$.

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

  • [Au] M. Auslander, Representation theory of Artin algebras. I, II. Comm. Algebra 1 (1974), 177-268; ibid. 1 (1974), 269-310. MR 0349747 (50:2240)
  • [BGS] A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010)
  • [BG] J. Bernstein and S. Gelfand, Tensor products of finite and infinite dimensional representations of semi-simple Lie algebras, Comp. Math. 41 (1980), 245-285. MR 581584 (82c:17003)
  • [BFK] J. Bernstein, I. B. Frenkel and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $ U(\mathfrak{sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no.2, 199-241. MR 1714141 (2000i:17009)
  • [BGG] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, A certain category of $ {\mathfrak{g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1-8. MR 0407097 (53:10880)
  • [CR] J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and $ sl_2$-categorification, to appear in Annals Math., math.RT/0407205.
  • [Do] J. M. 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 (98c:14042)
  • [FKS] I. B. Frenkel, M. Khovanov and C. Stroppel, A categorification of finite-dimensional irreducible representations of quantum $ sl(2)$ and their tensor products, Selecta Math. 12 (2006) nos. 3-4, 379-431. MR 2305608 (2008a:17014)
  • [Fu] 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. MR 1994220 (2004m:20087)
  • [Ir] R. S. Irving, Projective modules in the category $ \mathcal{O}_S$: self-duality, Trans. Amer. Math. Soc. 291 (1985), no. 2, 701-732. MR 800259 (87i:17005)
  • [Ja] J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, 750, Berlin, (1979). MR 552943 (81m:17011)
  • [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 560412 (81j:20066)
  • [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\sb n$, J. Algebra 173 (1995), no. 1, 97-121. MR 1327362 (96b:20013)
  • [Na] H. Naruse, On an isomorphism between Specht module and left cell of $ \mathfrak{S}\sb n,$ Tokyo J. Math. 12 (1989), no. 2, 247-267. MR 1030495 (90k:20025)
  • [R-C] A. Rocha-Caridi, Splitting criteria for $ \mathfrak{g}$-modules induced from a parabolic and the BGG resolution of a finite-dimensional, irreducible $ \mathfrak{g}$-module. Trans. Amer. Math. Soc. 262 (1980), no. 2, 335-366. MR 586721 (82f:17006)
  • [So] W. Soergel, Kazhdan-Lusztig polynomials and a combinatorics for tilting modules. Represent. Theory 1 (1997), 83-114. MR 1444322 (98d:17026)
  • [St1] C. Stroppel, Category $ \mathcal{O}$: gradings and translation functors, J. Algebra 268 (2003), no. 1, 301-326. MR 2005290 (2004i:17007)
  • [St2] C. Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, math.RT/0608234.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 17B10, 05E10, 20C08

Retrieve articles in all journals with MSC (2000): 17B10, 05E10, 20C08

Additional Information

Mikhail Khovanov
Affiliation: Department of Mathematics, Columbia University, New York, New York

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, Uppsala, Sweden

Catharina Stroppel
Affiliation: Department of Mathematics, University of Glasgow, Glasgow, United Kingdom

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

American Mathematical Society