Imprimitive irreducible modules for finite quasisimple groups / Gerhard Hiss, William J. Husen, Kay Magaard.

Format:

Book

Author/Creator:

Hiss, G., author.

Husen, William J., author.

Magaard, Kay, author.

Series:

Memoirs of the American Mathematical Society ; Volume 234, Number 1104 (fourth of 5 numbers)

Memoirs of the American Mathematical Society, 1947-6221 ; Volume 234, Number 1104 (fourth of 5 numbers)

Language:

English

Subjects (All):

Algebraic fields.

Finite groups.

Semisimple Lie groups.

Physical Description:

1 online resource (114 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2014.

Language Note:

English

Summary:

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 Harish-Chandra 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 Harish-Chandra philosophy to classify irreducible Harish-Chandra induced modules in terms of Harish-Chandra 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.

Contents:

""Cover""; ""Title page""; ""Acknowledgements""; ""Chapter 1. Introduction""; ""Chapter 2. Generalities""; ""2.1. Comments on the notation""; ""2.2. Conditions for primitivity""; ""2.3. Some results on linear groups of small degree""; ""2.4. Reduction modulo â?? and imprimitivity""; ""2.5. A result on polynomials""; ""Chapter 3. Sporadic Groups and the Tits Group""; ""Chapter 4. Alternating Groups""; ""Chapter 5. Exceptional Schur Multipliers and Exceptional Isomorphisms""; ""5.1. Description of the tables""; ""5.2. The proofs""

""9.3. Lusztig series""""9.4. Examples for the restriction to commutator subgroups""; ""Chapter 10. Exceptional groups""; ""10.1. The exceptional groups of type and ""; ""10.2. Explicit results on some exceptional groups""; ""Bibliography""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-2031-7