Cyclotomic -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

DOI:
https://doi.org/10.1090/S1088-4165-10-00375-4

Published electronically:
May 6, 2010

MathSciNet review:
2644457

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Abstract: Let be the Ariki-Koike algebra associated to the complex reflection group , and let be the cyclotomic -Schur algebra associated to , introduced by Dipper, James and Mathas. For each such that , we define a subalgebra of and its quotient algebra . It is shown that is a standardly based algebra and is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of , which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic -Schur algebras associated to Ariki-Koike algebras of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra of type , and prove the Schur-Weyl duality between and .

**[A]**Susumu Ariki,*Cyclotomic 𝑞-Schur algebras as quotients of quantum algebras*, J. Reine Angew. Math.**513**(1999), 53–69. MR**1713319**, https://doi.org/10.1515/crll.1999.063**[DJM]**Richard Dipper, Gordon James, and Andrew Mathas,*Cyclotomic 𝑞-Schur algebras*, Math. Z.**229**(1998), no. 3, 385–416. MR**1658581**, https://doi.org/10.1007/PL00004665**[DR]**Jie Du and Hebing Rui,*Based algebras and standard bases for quasi-hereditary algebras*, Trans. Amer. Math. Soc.**350**(1998), no. 8, 3207–3235. MR**1603902**, https://doi.org/10.1090/S0002-9947-98-02305-8**[GL]**J. J. Graham and G. I. Lehrer,*Cellular algebras*, Invent. Math.**123**(1996), no. 1, 1–34. MR**1376244**, https://doi.org/10.1007/BF01232365**[HS]**Jun Hu and Friederike Stoll,*On double centralizer properties between quantum groups and Ariki-Koike algebras*, J. Algebra**275**(2004), no. 1, 397–418. MR**2047454**, https://doi.org/10.1016/j.jalgebra.2003.10.026**[JM]**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**, https://doi.org/10.1090/S0002-9947-00-02492-2**[M]**Andrew Mathas,*The representation theory of the Ariki-Koike and cyclotomic 𝑞-Schur algebras*, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 261–320. MR**2074597****[Sa]**Nobuharu Sawada,*On decomposition numbers of the cyclotomic 𝑞-Schur algebras*, J. Algebra**311**(2007), no. 1, 147–177. MR**2309882**, https://doi.org/10.1016/j.jalgebra.2006.11.032**[Sh]**Toshiaki Shoji,*A Frobenius formula for the characters of Ariki-Koike algebras*, J. Algebra**226**(2000), no. 2, 818–856. MR**1752762**, https://doi.org/10.1006/jabr.1999.8178**[SakS]**Masahiro Sakamoto and Toshiaki Shoji,*Schur-Weyl reciprocity for Ariki-Koike algebras*, J. Algebra**221**(1999), no. 1, 293–314. MR**1722914**, https://doi.org/10.1006/jabr.1999.7973**[SawS]**Nobuharu Sawada and Toshiaki Shoji,*Modified Ariki-Koike algebras and cyclotomic 𝑞-Schur algebras*, Math. Z.**249**(2005), no. 4, 829–867. MR**2126219**, https://doi.org/10.1007/s00209-004-0739-8**[W]**K. Wada, On decomposition numbers with Jantzen filtration of cyclotomic -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

DOI:
https://doi.org/10.1090/S1088-4165-10-00375-4

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.