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THE INDUCTIVE BLOCKWISE ALPERIN WEIGHT CONDITION FOR TYPE ${\boldsymbol{\mathsf{C}}}$ AND THE PRIME $\textbf{2}$

Part of: Representation theory of groups

Published online by Cambridge University Press:  06 November 2020

ZHICHENG FENG  and
GUNTER MALLE
ZHICHENG FENG
Affiliation:
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China e-mail: zfeng@pku.edu.cn
GUNTER MALLE*
Affiliation:
FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
*
e-mail: malle@mathematik.uni-kl.de
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Abstract

We establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$. As a main step, we derive a labelling set for the irreducible $2$-Brauer characters of the finite symplectic groups $\operatorname {Sp}_{2n}(q)$ (with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.

Keywords

action of automorphisms on Brauer charactersAlperin weight conjecturesymplectic groupsunipotent classes

MSC classification

Primary: 20C20: Modular representations and characters
Secondary: 20C33: Representations of finite groups of Lie type

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 1 , August 2022 , pp. 1 - 20
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Anthony Henderson

The first author gratefully acknowledges financial support by NSFC (11631001 and 11901028) and Fundamental Research Funds for the Central Universities (No. FRF-TP-19-036A1). The second author gratefully acknowledges financial support by SFB TRR 195.

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