eBook ISBN:  9781470420314 
Product Code:  MEMO/234/1104.E 
List Price:  $80.00 
MAA Member Price:  $72.00 
AMS Member Price:  $48.00 
eBook ISBN:  9781470420314 
Product Code:  MEMO/234/1104.E 
List Price:  $80.00 
MAA Member Price:  $72.00 
AMS Member Price:  $48.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 234; 2014; 114 ppMSC: Primary 20
Motivated by the maximal subgroup problem of the finite classical groups the authors 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 \(G\).
The authors obtain their strongest results when \({\rm char}(K) = 0\), although much of their analysis carries over into positive characteristic. If \(G\) is a finite quasisimple group of Lie type, they prove that an imprimitive irreducible \(KG\)module is HarishChandra induced. This being true for \(\mbox{\rm char}(K)\) different from the defining characteristic of \(G\), the authors specialize to the case \({\rm char}(K) = 0\) and apply HarishChandra philosophy to classify irreducible HarishChandra induced modules in terms of HarishChandra series, as well as in terms of Lusztig series. The authors 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 their 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 \(G\), the authors determine all irreducible imprimitive \(KG\)modules for arbitrary characteristic of \(K\).

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 nonparabolic 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


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Motivated by the maximal subgroup problem of the finite classical groups the authors 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 \(G\).
The authors obtain their strongest results when \({\rm char}(K) = 0\), although much of their analysis carries over into positive characteristic. If \(G\) is a finite quasisimple group of Lie type, they prove that an imprimitive irreducible \(KG\)module is HarishChandra induced. This being true for \(\mbox{\rm char}(K)\) different from the defining characteristic of \(G\), the authors specialize to the case \({\rm char}(K) = 0\) and apply HarishChandra philosophy to classify irreducible HarishChandra induced modules in terms of HarishChandra series, as well as in terms of Lusztig series. The authors 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 their 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 \(G\), the authors determine all irreducible imprimitive \(KG\)modules for arbitrary characteristic of \(K\).

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 nonparabolic 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