Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

On decomposition numbers with Jantzen filtration of cyclotomic $ q$-Schur algebras


Author: Kentaro Wada
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
Published electronically: May 18, 2010
MathSciNet review: 2652073
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A1] S. Ariki.
    On the decomposition numbers of the Hecke algebra of $ G(m,1,n)$.
    J. Math. Kyoto Univ. 36 (1996), 789-808. MR 1443748 (98h:20012)
  • [A2] S. 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), 827-837. MR 1864465 (2002i:20004)
  • [AM] S. Ariki and A. Mathas.
    The number of simple modules of the Hecke algebras of type $ G(r,1,n)$.
    Math. Z. 233 (2000), 601-623. MR 1750939 (2001e:20007)
  • [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)
  • [Jac] N. Jacon.
    An algorithm for the computation of the decomposition matrices for Ariki-Koike algebras.
    J. Algebra 292 (2005), 100-109. MR 2166797 (2006g:20010)
  • [JM] G. 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)
  • [LLT] A. Lascoux, B. Leclerc, and J.-Y. Thibon.
    Hecke algebras at roots of unity and crystal bases of quantum affine algebras.
    Comm. Math. Phys. 181 (1996), 205-263. MR 1410572 (97k:17019)
  • [LT] B. Leclerc and J.-Y. Thibon.
    Canonical bases of $ q$-deformed Fock spaces.
    Internat. Math. Res. Notices, (1996) 447-456. MR 1399410 (97h:17023)
  • [M1] A. Mathas.
    Simple modules of Ariki-Koike algebras.
    In “ Group representations: cohomology, group actions and topology '', Proc. Sympos. Pure Math. vol. 63, Amer. Math. Soc., 1998, pp. 383-396. MR 1603195 (99d:20018)
  • [M2] A. Mathas.
    Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series Vol. 15,
    Amer. Math. Soc., 1999. MR 1711316 (2001g:20006)
  • [M3] 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. Pure Math. Vol. 40, Math. Soc. Japan, Tokyo 2004, pp. 261-320. MR 2074597 (2005f:20014)
  • [SW] T. Shoji and K. Wada.
    Cyclotomic $ q$-Schur algebras associated to the Ariki-Koike algebra, Represent. Theory 14 (2010), 379-416.
  • [U] D. Uglov.
    Canonical bases of higher-level $ q$-deformed Fock spaces and Kazhdan-Lusztig polynomials.
    In Physical combinatorics (Kyoto, 1999), Progr. Math. vol. 191, Birkhäuser Boston, Boston, 2000, pp. 249-299 MR 1768086 (2001k:17030)
  • [VV] M. Varagnolo and E. Vasserot.
    On the decomposition matrices of the quantized Schur algebra.
    Duke Math. J. 100, (1999), 267-297. MR 1722955 (2001c:17029)
  • [Y] X. Yvonne.
    A conjecture for $ q$-decomposition matrices of cyclotomic $ v$-Schur algebras.
    J. Algebra, 304, (2006) 419-456. MR 2256400 (2008d:16051)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 20-XX, 16-XX

Retrieve articles in all journals with MSC (2010): 20-XX, 16-XX


Additional 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

DOI: https://doi.org/10.1090/S1088-4165-2010-00376-3
Received by editor(s): November 6, 2007
Published electronically: May 18, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society