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On decomposition numbers and Alvis–Curtis duality

Published online by Cambridge University Press:  01 November 2007

BERND ACKERMANN  and
SIBYLLE SCHROLL
BERND ACKERMANN
Affiliation:
Inst.f.Algebra u.Zahlentheorie, Universität Stuttgart, 70550 Stuttgart, Germany. email: [email protected]
SIBYLLE SCHROLL*
Affiliation:
Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB. email: [email protected]
*
†The second author acknowledges support through a Marie Curie Fellowship.
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Abstract

We show that for general linear groups GLn(q) as well as for q-Schur algebras the knowledge of the modular Alvis–Curtis duality over fields of characteristic ℓ, ℓ ∤ q, is equivalent to the knowledge of the decomposition numbers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 3 , November 2007 , pp. 509 - 520
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Anderson, F. W. and Fuller, K. R.. Rings and categories of modules, second edition. Graduate Texts in Mathematics, 13 (Springer-Verlag, 1992).CrossRefGoogle Scholar
[2]Broué, M.. Isométries parfaites, types de blocs, catégories dérivées. Astérisque 181–182 (1990), 61–92.Google Scholar
[3]Brundan, J.. Modular branching rules and the Mullineux map for Hecke algebras of type A. Proc. London Math. Soc. (3) 77 (1998), no. 3, 551–581.CrossRefGoogle Scholar
[4]Brundan, J., Dipper, R. and Kleshchev, A.. Quantum linear groups and representations of Gl n(Fq). Mem. Amer. Math. Soc. 706.Google Scholar
[5]Carter, R.. Finite Groups of Lie Type (Wiley, 1985).Google Scholar
[6]Cabanes, M. and Rickard, J.. Alvis–Curtis duality as an equivalence of derived categories. Modular representation theory of finite groups (Charlottesville, VA, 1998), 157–174 (de Gruyter, 2001).CrossRefGoogle Scholar
[7]Dipper, R.. On the decomposition numbers of finite general linear groups II. Trans. Amer. Math. Soc. 292 (1985), 1, 123–133.CrossRefGoogle Scholar
[8]Dipper, R. and James, G.. Representations of Hecke algebras of general linear groups. Proc. London Math. Soc. (3) 52 (1986), no. 1, 20–52.CrossRefGoogle Scholar
[9]Dipper, R. and James, G.. The q-Schur algebra. Proc. London Math. Soc. (3) 59 (1989) 23–50.CrossRefGoogle Scholar
[10]Donkin, S.. The q-Schur Algebra. LMS Lecture Note Series 253 (Cambridge University Press, 1998).CrossRefGoogle Scholar
[11]Fayers, M.. Decomposition numbers for weight three blocks of symmetric groups and Iwahori–Hecke algebras. Trans. Amer. Math. Soc., to appear.Google Scholar
[12]Fong, P. and Srinivasan, B.. The blocks of finite general linear and unitary groups. Invent. Math. 69 (1982), 1, 109–153.CrossRefGoogle Scholar
[13]Green, J. A.. The characters of the finite general linear groups. Trans. Amer. Math. Soc. 80 (1955), 402–447.CrossRefGoogle Scholar
[14]Iwahori, N.. On the structure of a Hecke ring of a Chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215–236.Google Scholar
[15]James, G. D.. Representations of General Linear Groups. LMS Lecture Note Series 94 (Cambridge University Press, 1984).CrossRefGoogle Scholar
[16]James, G. D.. The irreducible representations of the finite general linear groups. Proc. London Math. Soc. (3) 52 (1986), 236–268.CrossRefGoogle Scholar
[17]James, G. D.. The decomposition matrices of GL n(q) for n ≤ 10. Proc. London Math. Soc. (3) 60 (1990), 225–265.CrossRefGoogle Scholar
[18]James, G. D., Lyle, S. and Mathas, A.. Rouquier blocks. Math. Z. 252 (2006), 511–531.CrossRefGoogle Scholar
[19]Linckelmann, M.. and Schroll, S.. A two-sided q-analogue of the Coxeter complex. J. Algebra (1) 289 (2005), 128–134.CrossRefGoogle Scholar
[20]Miyachi, H.. Some remarks on James' conjecture. Preprint (2001).Google Scholar
[21]Miyachi, H.. Unipotent block of finite general linear groups in non-defining characteristic, Ph.D. thesis. (Chiba University, 2001).Google Scholar
[22]Mullineux, G.. Bijections of p-regular partitions and p-modular irreducibles of the symmetric group. J. Algebra 34 (1979), 60–66.Google Scholar
[23]Rickard, J.. Derived equivalences as derived functors. J. London Math. Soc. (2) 43 (1991), no. 1, 37–48.CrossRefGoogle Scholar
[24]Schroll, S.. On the Alvis–Curtis duality and general linear groups. Ph. D. thesis. (Université Paris 13, 2003).Google Scholar
[25]Schroll, S.. Alvis–Curtis duality on q-Schur and Hecke algebras. Q. J. Math. 58 (2007), no. 2.CrossRefGoogle Scholar
[26]Takeuchi, M.. The group ring of GL sb n(q) and the q-Schur algebra. J. Math. Soc. Japan 48 (1996), 2, 259–274.CrossRefGoogle Scholar
[27]Tan, K. M.. Some decomposition numbers of q-Schur algebras. Preprint (2005).Google Scholar