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

HTML articles powered by AMS MathViewer

- by Jonathan Brundan
- Represent. Theory
**12**(2008), 236-259 - DOI: https://doi.org/10.1090/S1088-4165-08-00333-6
- Published electronically: July 29, 2008
- PDF | Request permission

## 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

- Tomoyuki Arakawa and Takeshi Suzuki,
*Duality between $\mathfrak {s}\mathfrak {l}_n(\textbf {C})$ and the degenerate affine Hecke algebra*, J. Algebra**209**(1998), no. 1, 288–304. MR**1652134**, DOI 10.1006/jabr.1998.7530 - Susumu Ariki, Andrew Mathas, and Hebing Rui,
*Cyclotomic Nazarov-Wenzl algebras*, Nagoya Math. J.**182**(2006), 47–134. MR**2235339**, DOI 10.1017/S0027763000026842 - Ken A. Brown and Ken R. Goodearl,
*Lectures on algebraic quantum groups*, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR**1898492**, DOI 10.1007/978-3-0348-8205-7 - Jonathan Brundan,
*Dual canonical bases and Kazhdan-Lusztig polynomials*, J. Algebra**306**(2006), no. 1, 17–46. MR**2271570**, DOI 10.1016/j.jalgebra.2006.01.053 - J. Brundan, Symmetric functions, parabolic category $\mathcal O$ and the Springer fiber,
*Duke Math. J.***143**(2008), 41–79. - Jonathan Brundan and Alexander Kleshchev,
*Hecke-Clifford superalgebras, crystals of type $A_{2l}^{(2)}$ and modular branching rules for $\hat S_n$*, Represent. Theory**5**(2001), 317–403. MR**1870595**, DOI 10.1090/S1088-4165-01-00123-6 - Jonathan Brundan and Alexander Kleshchev,
*Representation theory of symmetric groups and their double covers*, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 31–53. MR**1994959**, DOI 10.1142/9789812564481_{0}003 - J. Brundan and A. Kleshchev, Schur-Weyl duality for higher levels, to appear in
*Selecta Math.*; math.RT/0605217. - Roger W. Carter and George Lusztig,
*On the modular representations of the general linear and symmetric groups*, Math. Z.**136**(1974), 193–242. MR**354887**, DOI 10.1007/BF01214125 - V. G. Drinfel′d,
*Degenerate affine Hecke algebras and Yangians*, Funktsional. Anal. i Prilozhen.**20**(1986), no. 1, 69–70 (Russian). MR**831053** - David Eisenbud,
*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960**, DOI 10.1007/978-1-4612-5350-1 - Thomas J. Enright and Brad Shelton,
*Categories of highest weight modules: applications to classical Hermitian symmetric pairs*, Mem. Amer. Math. Soc.**67**(1987), no. 367, iv+94. MR**888703**, DOI 10.1090/memo/0367 - I. Grojnowski, Blocks of the cyclotomic Hecke algebra, preprint, 1999.
- 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 - G. D. James,
*The representation theory of the symmetric groups*, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR**513828**, DOI 10.1007/BFb0067708 - Gordon James and Andrew Mathas,
*The Jantzen sum formula for cyclotomic $q$-Schur algebras*, Trans. Amer. Math. Soc.**352**(2000), no. 11, 5381–5404. MR**1665333**, DOI 10.1090/S0002-9947-00-02492-2 - Jens C. Jantzen,
*Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren*, Math. Ann.**226**(1977), no. 1, 53–65. MR**439902**, DOI 10.1007/BF01391218 - A.-A. A. Jucys,
*Symmetric polynomials and the center of the symmetric group ring*, Rep. Mathematical Phys.**5**(1974), no. 1, 107–112. MR**419576**, DOI 10.1016/0034-4877(74)90019-6 - Mikhail Khovanov,
*Crossingless matchings and the cohomology of $(n,n)$ Springer varieties*, Commun. Contemp. Math.**6**(2004), no. 4, 561–577. MR**2078414**, DOI 10.1142/S0219199704001471 - Alexander Kleshchev,
*Linear and projective representations of symmetric groups*, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. MR**2165457**, DOI 10.1017/CBO9780511542800 - George Lusztig,
*Cuspidal local systems and graded Hecke algebras. I*, Inst. Hautes Études Sci. Publ. Math.**67**(1988), 145–202. MR**972345**, DOI 10.1007/BF02699129 - Sinéad Lyle and Andrew Mathas,
*Blocks of cyclotomic Hecke algebras*, Adv. Math.**216**(2007), no. 2, 854–878. MR**2351381**, DOI 10.1016/j.aim.2007.06.008 - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - Volodymyr Mazorchuk and Catharina Stroppel,
*Projective-injective modules, Serre functors and symmetric algebras*, J. Reine Angew. Math.**616**(2008), 131–165. MR**2369489**, DOI 10.1515/CRELLE.2008.020 - G. E. Murphy,
*The idempotents of the symmetric group and Nakayama’s conjecture*, J. Algebra**81**(1983), no. 1, 258–265. MR**696137**, DOI 10.1016/0021-8693(83)90219-3 - K. Platt,
*Classifying the representation type of infinitesimal blocks of category $\mathcal O_S$*, Ph.D. thesis, University of Georgia, 2008. - O. Ruff, Centers of cyclotomic Sergeev superalgebras, preprint, 2008.
- C. Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homolog, to appear in
*Compositio Math.*; math.RT/0608234. - Weiqiang Wang,
*Vertex algebras and the class algebras of wreath products*, Proc. London Math. Soc. (3)**88**(2004), no. 2, 381–404. MR**2032512**, DOI 10.1112/S0024611503014382

## Bibliographic Information

**Jonathan Brundan**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- Email: brundan@uoregon.edu
- Received by editor(s): August 15, 2006
- Received by editor(s) in revised form: June 25, 2008
- Published electronically: July 29, 2008
- Additional Notes: Research supported in part by NSF grant no. DMS-0654147.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**12**(2008), 236-259 - MSC (2000): Primary 20C08, 17B20
- DOI: https://doi.org/10.1090/S1088-4165-08-00333-6
- MathSciNet review: 2424964