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Cyclotomic $ q$-Schur algebras associated to the Ariki-Koike algebra

Authors: Toshiaki Shoji and Kentaro Wada
Journal: Represent. Theory 14 (2010), 379-416
MSC (2010): Primary 20C08, 20G43
Published electronically: May 6, 2010
MathSciNet review: 2644457
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Abstract: Let $ \mathcal{H}_{n,r}$ be the Ariki-Koike algebra associated to the complex reflection group $ \mathfrak{S}_n\ltimes (\mathbb{Z}/r\mathbb{Z})^n$, and let $ \mathcal{S}(\varLambda)$ be the cyclotomic $ q$-Schur algebra associated to $ \mathcal{H}_{n,r}$, introduced by Dipper, James and Mathas. For each $ \mathbf{p} = (r_1, \dots, r_g) \in \mathbb{Z}_{>0}^g$ such that $ r_1 +\cdots + r_g = r$, we define a subalgebra $ \mathcal{S}^{\mathbf{p}}$ of $ \mathcal{S}(\varLambda)$ and its quotient algebra $ \overline{\mathcal{S}}^{\mathbf{p}}$. It is shown that $ \mathcal{S}^{\mathbf{p}}$ is a standardly based algebra and $ \overline{\mathcal{S}}^{\mathbf{p}}$ is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of $ \mathcal{S}(\varLambda)$, which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic $ q$-Schur algebras associated to Ariki-Koike algebras $ \mathcal{H}_{n_i,r_i}$ of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra $ \overline{\mathcal{H}}^{\mathbf{p}}$ of type $ \mathbf{p}$, and prove the Schur-Weyl duality between $ \overline{\mathcal{H}}^{\mathbf{p}}$ and $ \overline{\mathcal{S}}^{\mathbf{p}}$.

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  • [A] S. Ariki, Cyclotomic $ q$-Schur algebras as quotients of quantum algebras, J. Reine Angew. Math. 513 (1999), 53-69. MR 1713319 (2001a:16065)
  • [DJM] R. Dipper, G. James and A. Mathas, Cyclotomic $ q$-Schur algebras, Math. Z. 229, (1998), 385-416. MR 1658581 (2000a:20033)
  • [DR] J. Du and H. Rui, Based algebras and standard bases for quasi-hereditary algebras, Trans. Amer. Math. Soc. 350 (1998), 3207-3235. MR 1603902 (99b:16027)
  • [GL] J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math., 123 (1996), 1-34. MR 1376244 (97h:20016)
  • [HS] J. Hu and F. Stoll, On double centralizer properties between quantum groups and Ariki-Koike algebras, J. Algebra 275 (2004), no. 1, 397-418. MR 2047454 (2005f:20012)
  • [JM] G.D. 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)
  • [M] A. Mathas, The representation theory of the Ariki-Koike and cyclotomic $ q$-Schur algebras, in ``Representation Theory of Algebraic Groups and Quantum Groups'', Adv. Stud. in Pure Math., 40 (2004), pp. 261-320. MR 2074597 (2005f:20014)
  • [Sa] N. Sawada, On decomposition numbers of the cyclotomic $ q$-Schur algebras, J. Algebra 311 (2007), no. 1, 147-177. MR 2309882 (2008c:20007)
  • [Sh] T. Shoji, A Frobenius formula for the characters of Ariki-Koike algebras. J. Algebra 226, (2000), 818-856. MR 1752762 (2001f:20013)
  • [SakS] M. Sakamoto and T. Shoji, Schur-Weyl reciprocity for Ariki-Koike algebras, J. Algebra 221 (1999), 293-314. MR 1722914 (2001f:17030)
  • [SawS] N. Sawada and T. Shoji, Modified Ariki-Koike algebras and cyclotomic $ q$-Schur algebras, Math. Z. 249 (2005), 829-867. MR 2126219 (2005j:20006)
  • [W] K. Wada, On decomposition numbers with Jantzen filtration of cyclotomic $ q$-Schur algebras, to appear in Representation Theory, 2010.

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Additional Information

Toshiaki Shoji
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Kentaro Wada
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
Address at time of publication: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 Japan

Received by editor(s): November 1, 2007
Received by editor(s) in revised form: February 6, 2010
Published electronically: May 6, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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