A categorification of integral Specht modules

Khovanov, Mikhail; Mazorchuk, Volodymyr; Stroppel, Catharina

doi:10.1090/S0002-9939-07-09124-1

A categorification of integral Specht modules
HTML articles powered by AMS MathViewer

by Mikhail Khovanov, Volodymyr Mazorchuk and Catharina Stroppel PDF

Proc. Amer. Math. Soc. 136 (2008), 1163-1169 Request permission

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

Maurice Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310. MR 349747, DOI 10.1080/00927877408548230
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, DOI 10.1090/S0894-0347-96-00192-0
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
Joseph Bernstein, Igor Frenkel, and Mikhail 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, DOI 10.1007/s000290050047
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak {g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and $sl_2$-categorification, to appear in Annals Math., math.RT/0407205.
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, DOI 10.1006/aama.1996.0015
Igor Frenkel, Mikhail Khovanov, and Catharina Stroppel, A categorification of finite-dimensional irreducible representations of quantum $\mathfrak {sl}_2$ and their tensor products, Selecta Math. (N.S.) 12 (2006), no. 3-4, 379–431. MR 2305608, DOI 10.1007/s00029-007-0031-y
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, DOI 10.1016/S0001-8708(02)00072-5
Ronald S. Irving, Projective modules in the category ${\scr O}_S$: self-duality, Trans. Amer. Math. Soc. 291 (1985), no. 2, 701–732. MR 800259, DOI 10.1090/S0002-9947-1985-0800259-9
Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
V. Mazorchuk and C. Stroppel, Projective-injective modules, Serre functors and symmetric algebras, to appear in J. reine angew. Math. math.RT/0508119.
G. E. Murphy, The representations of Hecke algebras of type $A_n$, J. Algebra 173 (1995), no. 1, 97–121. MR 1327362, DOI 10.1006/jabr.1995.1079
Hiroshi Naruse, On an isomorphism between Specht module and left cell of $\mathfrak {S}_n$, Tokyo J. Math. 12 (1989), no. 2, 247–267. MR 1030495, DOI 10.3836/tjm/1270133181
Alvany Rocha-Caridi, Splitting criteria for ${\mathfrak {g}}$-modules induced from a parabolic and the Berňsteĭn-Gel′fand-Gel′fand resolution of a finite-dimensional, irreducible ${\mathfrak {g}}$-module, Trans. Amer. Math. Soc. 262 (1980), no. 2, 335–366. MR 586721, DOI 10.1090/S0002-9947-1980-0586721-0
Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83–114. MR 1444322, DOI 10.1090/S1088-4165-97-00021-6
Catharina Stroppel, Category ${\scr O}$: gradings and translation functors, J. Algebra 268 (2003), no. 1, 301–326. MR 2005290, DOI 10.1016/S0021-8693(03)00308-9
C. Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, math.RT/0608234.