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

HTML articles powered by AMS MathViewer

- by Toshiaki Shoji and Kentaro Wada
- Represent. Theory
**14**(2010), 379-416 - DOI: https://doi.org/10.1090/S1088-4165-10-00375-4
- Published electronically: May 6, 2010
- PDF | Request permission

## 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 $\overline {\mathcal {S}}^{\mathbf {p}}$.## References

- Susumu Ariki,
*Cyclotomic $q$-Schur algebras as quotients of quantum algebras*, J. Reine Angew. Math.**513**(1999), 53–69. MR**1713319**, DOI 10.1515/crll.1999.063 - Richard Dipper, Gordon James, and Andrew Mathas,
*Cyclotomic $q$-Schur algebras*, Math. Z.**229**(1998), no. 3, 385–416. MR**1658581**, DOI 10.1007/PL00004665 - 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**, DOI 10.1090/S0002-9947-98-02305-8 - J. J. Graham and G. I. Lehrer,
*Cellular algebras*, Invent. Math.**123**(1996), no. 1, 1–34. MR**1376244**, DOI 10.1007/BF01232365 - 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**, DOI 10.1016/j.jalgebra.2003.10.026 - 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 - 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**, DOI 10.2969/aspm/04010261 - Nobuharu Sawada,
*On decomposition numbers of the cyclotomic $q$-Schur algebras*, J. Algebra**311**(2007), no. 1, 147–177. MR**2309882**, DOI 10.1016/j.jalgebra.2006.11.032 - Toshiaki Shoji,
*A Frobenius formula for the characters of Ariki-Koike algebras*, J. Algebra**226**(2000), no. 2, 818–856. MR**1752762**, DOI 10.1006/jabr.1999.8178 - Masahiro Sakamoto and Toshiaki Shoji,
*Schur-Weyl reciprocity for Ariki-Koike algebras*, J. Algebra**221**(1999), no. 1, 293–314. MR**1722914**, DOI 10.1006/jabr.1999.7973 - Nobuharu Sawada and Toshiaki Shoji,
*Modified Ariki-Koike algebras and cyclotomic $q$-Schur algebras*, Math. Z.**249**(2005), no. 4, 829–867. MR**2126219**, DOI 10.1007/s00209-004-0739-8 - K. Wada, On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras, to appear in Representation Theory, 2010.

## Bibliographic 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
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**14**(2010), 379-416 - MSC (2010): Primary 20C08, 20G43
- DOI: https://doi.org/10.1090/S1088-4165-10-00375-4
- MathSciNet review: 2644457