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Imprimitive irreducible modules for finite quasisimple groups

About this Title

Gerhard Hiss, Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany, William J. Husen, Department of Mathematics, The Ohio State University, Ohio 43210-1174 and Kay Magaard, School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 234, Number 1104
ISBNs: 978-1-4704-0960-9 (print); 978-1-4704-2031-4 (online)
DOI: https://doi.org/10.1090/memo/1104
Published electronically: August 1, 2014
Keywords: Finite quasisimple group, maximal subgroup, finite classical group, $\mathcal {C}_2$-subgroup, imprimitive representation
MSC: Primary 20B15, 20C33, 20C34, 20E28; Secondary 20B25, 20C15, 20C20

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Table of Contents

Chapters

  • Acknowledgements
  • 1. Introduction
  • 2. Generalities
  • 3. Sporadic Groups and the Tits Group
  • 4. Alternating Groups
  • 5. Exceptional Schur Multipliers and Exceptional Isomorphisms
  • 6. Groups of Lie type: Induction from non-parabolic subgroups
  • 7. Groups of Lie type: Induction from parabolic subgroups
  • 8. Groups of Lie type: char$(K) = 0$
  • 9. Classical groups: $\text {char}(K) = 0$
  • 10. Exceptional groups

Abstract

Motivated by the maximal subgroup problem of the finite classical groups we begin the classification of imprimitive irreducible modules of finite quasisimple groups over algebraically closed fields $K$. A module of a group $G$ over $K$ is imprimitive, if it is induced from a module of a proper subgroup of $G$.

We obtain our strongest results when char$(K) = 0$, although much of our analysis carries over into positive characteristic. If $G$ is a finite quasisimple group of Lie type, we prove that an imprimitive irreducible $KG$-module is Harish-Chandra induced. This being true for char$(K)$ different from the defining characteristic of $G$, we specialize to the case char$(K) = 0$ and apply Harish-Chandra philosophy to classify irreducible Harish-Chandra induced modules in terms of Harish-Chandra series, as well as in terms of Lusztig series. We determine the asymptotic proportion of the irreducible imprimitive $KG$-modules, when $G$ runs through a series groups of fixed (twisted) Lie type. One of the surprising outcomes of our investigations is the fact that these proportions tend to $1$, if the Lie rank of the groups tends to infinity.

For exceptional groups $G$ of Lie type of small rank, and for sporadic groups $G$, we determine all irreducible imprimitive $KG$-modules for arbitrary characteristic of $K$.

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References

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