Decomposition numbers for Hecke algebras of type G (r , p , n ): the (ε, q )-separated case

DOI10.1112/PLMS/PDR047zbMATH Open1254.20008arXiv1004.3928OpenAlexW2083604144MaRDI QIDQ2888908

Andrew Mathas, Hu Jun

Publication date: 4 June 2012

Published in: Proceedings of the London Mathematical Society (Search for Journal in Brave)

Abstract: The paper studies the modular representation theory of the cyclotomic Hecke algebras of type $G(r,p,n)$ with $(eps,q)$-separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type $G(s,1,m)$, where $1lesler$ and $1lemlen$. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers. Consequently, in principle, the decomposition matrices of these algebras are now known in characteristic zero. In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type $G(r,p,n)$ when the parameters are $(eps,q)$-separated. The main results of the paper rest upon two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the extit{$l$-splittable decomposition numbers} and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type $G(r,p,n)$.

Full work available at URL: https://arxiv.org/abs/1004.3928

zbMATH Keywords

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Mathematics Subject Classification ID

Combinatorial aspects of representation theory (05E10) Hecke algebras and their representations (20C08) Representations of finite symmetric groups (20C30) Modular representations and characters (20C20)

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