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 | References | Similar Articles | Additional Information

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]**S. Ariki, Cyclotomic -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 -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 -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 -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 -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 -Schur algebras, Math. Z.**249**(2005), 829-867. MR**2126219 (2005j:20006)****[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.