Projective representations and spin characters of complex reflection groups \(G(m,p,n)\) and \(G(m,p,\infty)\). (English) Zbl 1270.20008

Hirai, Takeshi; Hora, Akihito; Hirai, Etsuko

MSJ Memoirs 29. Tokyo: Mathematical Society of Japan (ISBN 978-4-86497-017-4/pbk). x, 272 p. (2013).

Preface: This volume consists of one expository paper and two research papers, as follows:
1. T. Hirai, A. Hora and E. Hirai, Introductory expositions on projective representations of groups (referred as [E]);
2. T. Hirai, E. Hirai and A. Hora, Projective representations and spin characters of complex reflection groups \(G(m,p,n)\) and \(G(m,p,\infty)\). I. (referred as [I]);
T. Hirai, A. Hora and E. Hirai, Projective representations and spin characters of complex reflection groups \(G(m,p,n)\) and \(G(m,p,\infty)\). II: Case of generalized symmetric groups. (referred as [II]).
Since Schur’s trilogy of 1904, 1907 and 1911, many mathematicians studied projective representations of groups and algebras, and also of their characters. Nevertheless an introductory and expository article will serve to invite mathematicians to this interesting and important area. In this connection, the first paper [E] collects introductory expositions, with a historical plotting, for the theory of projective representations of groups and their characters.
The second paper [I] treats general theory for projective (or spin) representations and spin characters of complex reflection groups \(G(m,p,n)\) and \(G(m,p,\infty)=\lim_{n\to\infty}G(m,p,n)\), and clarifies the intimate relations between mother groups \(G(m,1,n)\) and \(G(m,1,\infty)\), \(p=1\), called generalized symmetric groups, and their child groups, \(G(m,p,n)\), \(G(m,p,\infty)\), \(p\mid m\), \(p>1\).
Also in the last one-third of the paper, we treat explicitly a case of spin type in connection with the case of non-spin type (i.e., the case of linear representations).
The third paper [II] treats spin irreducible representations and spin characters of generalized symmetric groups (mother groups) for two other spin types.
Each paper has its own abstract and introduction. References for research papers [I] and [II] are collected at the end of [II], whereas those for paper [E] remain at the end of [E] for convenience of readers, since they are chosen from historical and/or expository points of view and need not mixed up with many others for [I] and [II], very much specialized.
The articles of this volume will not be indexed individually.

MSC:

20C25	Projective representations and multipliers
20F55	Reflection and Coxeter groups (group-theoretic aspects)
20-02	Research exposition (monographs, survey articles) pertaining to group theory
20-06	Proceedings, conferences, collections, etc. pertaining to group theory
00B15	Collections of articles of miscellaneous specific interest
20C30	Representations of finite symmetric groups
05E05	Symmetric functions and generalizations
20C32	Representations of infinite symmetric groups
05E10	Combinatorial aspects of representation theory
22D12	Other representations of locally compact groups

Keywords:

projective representations; spin irreducible representations; spin characters; complex reflection groups; generalized symmetric groups

Citations:

JFM 35.0155.01; JFM 38.0174.02; JFM 41.0171.01