Representation Theory of Finite Group Extensions: Clifford Theory, Mackey Obstruction, and the Orbit Method

This monograph discusses the irreducible representations of finite groups using an operational and functional analytic approach,with a focus on the Little Group Method and Kirillov's Orbit Method. It also introduces projective representations to overcome the Mackey obstruction and applies the theory to various finite groups,including alternating,metacyclic,quaternionic,dihedral,and Heisenberg groups.

Format: Hardback
Length: 340 pages
Publication date: 30 November 2022
Publisher: Springer International Publishing AG

This monograph delves into the intricate analysis of a specific problem in group theory. It adopts an operational and functional analytic approach, seeking to describe the irreducible representations of a given finite group, G, through the structure of a short exact sequence 1 → N → G → H → 1. This problem has garnered significant attention from mathematicians such as I. Schur, A.H. Clifford, and G. Mackey, as well as more recent scholars like M. Isaacs, B. Huppert, Y.G. Berkovich, and E.M. Zhmud, and J.M.G. Fell and R.S. Doran.

The central themes of this monograph revolve around Clifford Theory and the Little Group Method (developed by Mackey and Wigner) for induced representations. On the other hand, Kirillov's Orbit Method, particularly applicable to step-2 nilpotent groups of odd order, establishes a profound connection between Lie rings and nilpotent groups. This method offers a natural and powerful framework for understanding the representation theory of various groups.

As an application, the monograph provides a detailed description of the representation theory of alternating groups, metacyclic, quaternionic, dihedral groups, and the finite Heisenberg group. The Little Group Method can be employed if a suitable unitary 2-cocycle, known as the Mackey obstruction, is trivial. To overcome this obstacle, unitary projective representations are introduced, leading to the development of corresponding Mackey and Clifford theories. Additionally, the commutant of an induced representation and the relative Hecke algebra are explored in depth.

Furthermore, the monograph presents a comprehensive exposition of the theory of projective representations for finite Abelian groups. This theory is applied to obtain a complete description of the irreducible representations of finite metabelian groups of odd order. The insights gained from this study shed light on the rich structure and properties of these groups, contributing to the ongoing development of group theory.

In summary, this monograph offers a comprehensive and detailed exploration of the problem of describing the irreducible representations of finite groups through the lens of operational and functional analytic techniques. It encompasses various topics, including Clifford Theory, the Little Group Method, Kirillov's Orbit Method, projective representations, and the representation theory of various groups. By applying these methods, the monograph provides valuable insights into the complex world of group theory, making it an essential resource for mathematicians and scholars interested in this field.

Weight: 699g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031138720
Edition number: 1st ed. 2022