On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras
HTML articles powered by AMS MathViewer
- by Kentaro Wada
- Represent. Theory 14 (2010), 417-434
- DOI: https://doi.org/10.1090/S1088-4165-2010-00376-3
- Published electronically: May 18, 2010
- PDF | Request permission
Abstract:
Let $\mathscr {S}(\varLambda )$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\mathscr {H}_{n,r}$, introduced by Dipper, James and Mathas. In this paper, we consider $v$-decomposition numbers of $\mathscr {S}(\varLambda )$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji and Wada, a product formula for $v$-decomposition numbers of $\mathscr {S}(\varLambda )$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $\mathscr {H}_{n_i,r_i}$ of smaller rank. Moreover, we prove a similar formula for $v$-decomposition numbers of $\mathscr {H}_{n,r}$ by using a Schur functor.References
- Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no.Β 4, 789β808. MR 1443748, DOI 10.1215/kjm/1250518452
- Susumu Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type $G(m,1,n)$ and Kleshchev multipartitions, Osaka J. Math. 38 (2001), no.Β 4, 827β837. MR 1864465
- Susumu Ariki and Andrew Mathas, The number of simple modules of the Hecke algebras of type $G(r,1,n)$, Math. Z. 233 (2000), no.Β 3, 601β623. MR 1750939, DOI 10.1007/s002090050489
- Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic $q$-Schur algebras, Math. Z. 229 (1998), no.Β 3, 385β416. 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.Β 8, 3207β3235. MR 1603902, DOI 10.1090/S0002-9947-98-02305-8
- J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no.Β 1, 1β34. MR 1376244, DOI 10.1007/BF01232365
- Nicolas Jacon, An algorithm for the computation of the decomposition matrices for Ariki-Koike algebras, J. Algebra 292 (2005), no.Β 1, 100β109. MR 2166797, DOI 10.1016/j.jalgebra.2004.10.017
- Gordon James and Andrew Mathas, The Jantzen sum formula for cyclotomic $q$-Schur algebras, Trans. Amer. Math. Soc. 352 (2000), no.Β 11, 5381β5404. MR 1665333, DOI 10.1090/S0002-9947-00-02492-2
- Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no.Β 1, 205β263. MR 1410572, DOI 10.1007/BF02101678
- Bernard Leclerc and Jean-Yves Thibon, Canonical bases of $q$-deformed Fock spaces, Internat. Math. Res. Notices 9 (1996), 447β456. MR 1399410, DOI 10.1155/S1073792896000293
- Andrew Mathas, Simple modules of Ariki-Koike algebras, Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp.Β 383β396. MR 1603195, DOI 10.1090/pspum/063/1603195
- Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316, DOI 10.1090/ulect/015
- 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.Β 261β320. MR 2074597, DOI 10.2969/aspm/04010261
- T. Shoji and K. Wada. Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra, Represent. Theory 14 (2010), 379-416.
- Denis Uglov, Canonical bases of higher-level $q$-deformed Fock spaces and Kazhdan-Lusztig polynomials, Physical combinatorics (Kyoto, 1999) Progr. Math., vol. 191, BirkhΓ€user Boston, Boston, MA, 2000, pp.Β 249β299. MR 1768086
- Michela Varagnolo and Eric Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), no.Β 2, 267β297. MR 1722955, DOI 10.1215/S0012-7094-99-10010-X
- Xavier Yvonne, A conjecture for $q$-decomposition matrices of cyclotomic $v$-Schur algebras, J. Algebra 304 (2006), no.Β 1, 419β456. MR 2256400, DOI 10.1016/j.jalgebra.2006.03.048
Bibliographic Information
- 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
- Email: wada@kurims.kyoto-u.ac.jp
- Received by editor(s): November 6, 2007
- Published electronically: May 18, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 14 (2010), 417-434
- MSC (2010): Primary 20-XX, 16-XX
- DOI: https://doi.org/10.1090/S1088-4165-2010-00376-3
- MathSciNet review: 2652073