Representation theory of finite group extensions : Cclifford theory, Mackey obstruction, and the orbit method / Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli.
Ceccherini-Silberstein, Tullio.; Scarabotti, Fabio.; Tolli, Filippo, 1968-
2022
QA176
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Title Representation theory of finite group extensions : Cclifford theory, Mackey obstruction, and the orbit method / Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli.
Author Ceccherini-Silberstein, Tullio.
ISBN 9783031138737 (electronic bk.)
3031138732 (electronic bk.)
3031138724
9783031138720
DOI https://doi.org/10.1007/978-3-031-13873-7
Imprint Cham, Switzerland : Springer, 2022.
Language English
Description 1 online resource.
Other Standard Identifiers 10.1007/978-3-031-13873-7 doi
Call Number QA176
Dewey Decimal Classification 512/.22
Summary This monograph adopts an operational and functional analytic approach to the following problem: given a short exact sequence (group extension) 1 N G H 1 of finite groups, describe the irreducible representations of G by means of the structure of the group extension. This problem has attracted many mathematicians, including I. Schur, A.H. Clifford, and G. Mackey and, more recently, M. Isaacs, B. Huppert, Y.G. Berkovich & E.M. Zhmud, and J.M.G. Fell & R.S. Doran. The main topics are, on the one hand, Clifford Theory and the Little Group Method (of Mackey and Wigner) for induced representations, and, on the other hand, Kirillovs Orbit Method (for step-2 nilpotent groups of odd order) which establishes a natural and powerful correspondence between Lie rings and nilpotent groups. As an application, a detailed description is given of the representation theory of the alternating groups, of metacyclic, quaternionic, dihedral groups, and of the (finite) Heisenberg group. The Little Group Method may be applied if and only if a suitable unitary 2-cocycle (the Mackey obstruction) is trivial. To overcome this obstacle, (unitary) projective representations are introduced and corresponding Mackey and Clifford theories are developed. The commutant of an induced representation and the relative Hecke algebra is also examined. Finally, there is a comprehensive exposition of the theory of projective representations for finite Abelian groups which is applied to obtain a complete description of the irreducible representations of finite metabelian groups of odd order.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (Springer, viewed December 8, 2022).
Added Author Scarabotti, Fabio.
Tolli, Filippo, 1968-
Series Springer monographs in mathematics.
Available in Other Form Print version: 9783031138720
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2.5 Some Applications of Mackey Theory to Clifford Theory
2.6 The G-Action on the N-Conjugacy Classes
2.7 Real, Complex, and Quaternionic Representations and Clifford Theory
2.8 Semidirect Products with an Abelian Normal Subgroup
2.9 Semidirect Products of Abelian Groups
2.10 Representation Theory of Wreath Products of Finite Groups
2.11 Multiplicity-Free Normal Subgroups
3 Abelian Extensions
3.1 The Dual Action
3.2 The Conjugation Action
3.3 The Intermediary Representations
3.4 Diagrammatic Summaries
4 The Little Group Method for Abelian Extensions

4.1 General Theory
4.2 Normal Subgroups with the Prime Condition
4.3 Normal Subgroups of Prime Index
4.4 The Case of Index Two Subgroups
5 Examples and Applications
5.1 Representation Theory and Conjugacy Classes of the Symmetric Groups Sn
5.2 Conjugacy Classes of An
5.3 The Irreducible Representations of An
5.4 Ambivalence of the Groups An
5.5 An Application to Isaacs' Going Down Theorem
5.6 Another Application: Analysis of p2-Extensions
5.7 Representation Theory of Finite Metacyclic Groups
5.8 Examples: Dihedral and Generalized Quaternion Groups

6 Central Extensions and the Orbit Method
6.1 Central Extensions
6.2 2-Divisible Abelian Groups, Equalized Cocycles, and Schur Multipliers
6.3 Lie Rings
6.4 The Cocycle Decomposition
6.5 The Malcev Correspondence
6.6 The Orbit Method
6.7 More on the Orbit Method: Induced Representations
6.8 More on the Orbit Method: Restricting to a Subgroup
6.9 The Orbit Method for the Finite Heisenberg Group
6.10 Restricting from Hqt to Hq
6.11 The Little Group Method for the Heisenberg Group
7 Representations of Finite Group Extensions via Projective Representations

7.1 Mackey Obstruction
7.2 Unitary Projective Representations
7.3 The Dual of a Group Extension
7.4 Central Extensions and the Finite Heisenberg Group
7.5 Analysis of the Commutant
7.6 The Hecke Algebra
8 Induced Projective Representations
8.1 Basic Theory
8.2 Mackey's Theory for Induced Projective Representations
9 Clifford Theory for Projective Representations
9.1 Preliminaries and Notation
9.2 Basic Clifford Theory for Projective Representations
9.3 Projective Unitary Representations of a Group Extension

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