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 Representation Theory
Representation Theory
ISSN 1088-4165

     

Centers of degenerate cyclotomic Hecke algebras and parabolic category $ \mathcal O$

Author(s): Jonathan Brundan
Journal: Represent. Theory 12 (2008), 236-259.
MSC (2000): Primary 20C08, 17B20
Posted: July 29, 2008
MathSciNet review: 2424964
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Abstract | References | Similar articles | Additional information

Abstract: We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type $ B_d(l)$ consists of symmetric polynomials in its commuting generators. The classification of the blocks of the degenerate cyclotomic Hecke algebras is an easy consequence. We then apply Schur-Weyl duality for higher levels to deduce analogous results for parabolic category $ \mathcal O$ for the Lie algebra $ \mathfrak{gl}_n(\mathbb{C})$.

References:

[AS]
T. Arakawa and T. Suzuki, Duality between $ \mathfrak{sl}_n(\mathbb{C})$ and the degenerate affine Hecke algebra, J. Algebra 209 (1998), 288-304. MR 1652134 (99h:17005)

[AMR]
S. Ariki, A. Mathas and H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006), 47-134. MR 2235339 (2007d:20005)

[BG]
K. Brown and K. Goodearl, Lectures on algebraic quantum groups, Birkhäuser, 2002. MR 1898492 (2003f:16067)

[B1]
J. Brundan, Dual canonical bases and Kazhdan-Lusztig polynomials, J. Algebra 306 (2006), 17-46. MR 2271570 (2007m:05229)

[B2]
J. Brundan, Symmetric functions, parabolic category $ \mathcal O$ and the Springer fiber, Duke Math. J. 143 (2008), 41-79.

[BK1]
J. Brundan and A. Kleshchev, Hecke-Clifford superalgebras, crystals of type $ A_{2\ell}^{(2)}$ and modular branching rules for $ \widehat{S}_n$, Represent. Theory 5 (2001), 317-403. MR 1870595 (2002j:17024)

[BK2]
J. Brundan and A. Kleshchev, Representation theory of symmetric groups and their double covers, in: Groups, combinatorics and geometry (Durham, 2001), pp. 31-53, World Scientific, 2003. MR 1994959 (2004i:20016)

[BK3]
J. Brundan and A. Kleshchev, Schur-Weyl duality for higher levels, to appear in Selecta Math.; math.RT/0605217.

[CL]
R. Carter and G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193-242. MR 0354887 (50:7364)

[D]
V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Func. Anal. Appl. 20 (1986), 56-58. MR 831053 (87m:22044)

[E]
D. Eisenbud, Commutative algebra with a view towards algebraic geometry, GTM 150, Springer, 1994. MR 1322960 (97a:13001)

[ES]
T. Enright and B. Shelton, Categories of highest weight modules: Applications to classical Hermitian symmetric pairs, Mem. Amer. Math. Soc. 67 (1987), no. 367. MR 888703 (88f:22052)

[G]
I. Grojnowski, Blocks of the cyclotomic Hecke algebra, preprint, 1999.

[I]
R. Irving, Projective modules in the category $ \mathcal O_S$: Self-duality, Trans. Amer. Math. Soc. 291 (1985), 701-732. MR 800259 (87i:17005)

[J]
G. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, no. 682, Springer, 1978. MR 513828 (80g:20019)

[JM]
G. James and A. Mathas, The Jantzen sum formula for cyclotomic $ q$-Schur algebras. Trans. Amer. Math. Soc. 352 (2000), 5381-5404. MR 1665333 (2001b:16017)

[Ja]
J. Jantzen, Kontravariante Formen aud induzierten Darstellungen halbeinfachen Lie-Algebren, Math. Ann. 226 (1977), 53-65. MR 0439902 (55:12783)

[Ju]
A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Report Math. Phys. 5 (1974), 107-112. MR 0419576 (54:7597)

[Kh]
M. Khovanov, Crossingless matchings and the cohomology of $ (n,n)$ Springer varieties, Commun. Contemp. Math. 6 (2004), 561-577. MR 2078414 (2005g:14090)

[K]
A. Kleshchev, Linear and Projective Representations of Symmetric Groups, Cambridge University Press, Cambridge, 2005. MR 2165457 (2007b:20022)

[L]
G. Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145-202. MR 972345 (90e:22029)

[LM]
S. Lyle and A. Mathas, Blocks of cyclotomic Hecke algebras, Adv. Math 216 (2007), 854-878. MR 2351381

[Mac]
I. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, second edition, OUP, 1995. MR 1354144 (96h:05207)

[MS]
V. Mazorchuk and C. Stroppel, Projective-injective modules, Serre functors and symmetric algebras, J. Reine Angew. Math. 616 (2008), 131-165. MR 2369489

[M]
G. Murphy, The idempotents of the symmetric group and Nakayama's conjecture, J. Algebra 81 (1983), 258-265. MR 696137 (84k:20007)

[P]
K. Platt, Classifying the representation type of infinitesimal blocks of category $ \mathcal O_S$, Ph.D. thesis, University of Georgia, 2008.

[R]
O. Ruff, Centers of cyclotomic Sergeev superalgebras, preprint, 2008.

[S]
C. Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homolog, to appear in Compositio Math.; math.RT/0608234.

[W]
W. Wang, Vertex algebras and the class algebras of wreath products, Proc. London Math. Soc. 88 (2004), 381-404. MR 2032512 (2005a:17029)

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Additional Information:

Jonathan Brundan
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: brundan@uoregon.edu

DOI: 10.1090/S1088-4165-08-00333-6
PII: S 1088-4165(08)00333-6
Received by editor(s): August 15, 2006
Received by editor(s) in revised form: June 25, 2008
Posted: July 29, 2008
Additional Notes: Research supported in part by NSF grant no. DMS-0654147.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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