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, Kirillov’s 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.
Author(s): Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Series: Springer Monographs in Mathematics
Publisher: Springer
Year: 2022
Language: English
Pages: 346
City: Cham
Preface
Contents
1 Preliminaries
1.1 Representations of Finite Groups
1.2 The Group Algebra and the Left-Regular Representation
1.3 Induced Representations
1.4 Further Results on Induced Representations
1.5 Semidirect Products, Wreath Products, and Group Extensions
1.6 Regular Wreath Products and the Kaloujnine-KrasnerTheorem
2 Clifford Theory
2.1 Preliminaries and Notation
2.2 Basic Clifford Theory
2.3 First Applications and the Little Group Method
2.4 The Case Where AG(σ) -1.2mu=IG(σ)/N is Abelian
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
10 Projective Representations of Finite Abelian Groups with Applications
10.1 Bicharacters and 2-Cocycles on Finite Abelian Groups
10.2 The Irreducible Projective Representations of Finite Abelian Groups
10.3 Representation Theory of Finite Metabelian Groups
10.4 Representation Theory of Finite Step-2 Nilpotent Groups
A Notes
A.1 Group Extensions and Cohomology
A.2 Clifford Theory
A.3 The Little Group Method and Its Applications
A.4 Lie Rings and the Orbit Method
A.5 Projective Representations
References
Subject index
Index of authors
