Representation theory of finite groups: algebra and arithmetic. (English) Zbl 1078.20009

Weintraub, Steven H.

Graduate Studies in Mathematics 59. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3222-0/hbk). ix, 212 p. (2003).

The origin of group representation theory can be traced to Frobenius’s paper of 1896, on the factorization of the group determinant. Of course, aspects of group representation theory were already present in the number-theoretic work of Dirichlet and Dedekind, who made critical use of the characters of finite Abelian groups. Following Frobenius’s somewhat complicated presentation of this non-Abelian generalization of the circulant, the concepts of group characters, with their attendant orthogonality relations, and of the representation of a group by linear transformations emerged to clarify the nature of the subject-matter Frobenius had introduced. By the time of the publication of the second edition of Burnside’s The Theory of Groups of Finite Order in 1911, the combined contributions of Frobenius, Schur and Burnside himself had shown the power of the method of group characters and representations to unravel the structure of abstract groups, as well as to illuminate aspects of algebraic geometry and the theory of symmetric functions. Following the first phase of the subject, Richard Brauer subsequently deepened it by relating the study of the group algebra to the theory of algebras, especially division algebras, and by introducing the modular theory of group representations. By the end of the 20th century, group representation theory had become a central method of algebra, with major applications in the theory of finite groups of Lie type, permutation groups, combinatorial theory and Galois theory. The theory concerns both finite groups and infinite groups, where remarkable analogies exist between the representations of compact Lie groups and their finite analogues.
Many books have been written on the characters and representations of finite groups, including the classics by Curtis and Reiner (1962), Feit (1967) and Isaacs (1976). Books such as these, with their wealth of applications, rich collection of exercises and depth of coverage, can be hard acts to follow, leaving very little that has not been said at the comparatively elementary or non-specialized level. The author of the book under review, Steven Weintraub, has entered the demanding arena of providing another contribution to the exposition of representation theory. We must confess to being a little disappointed by Weintraub’s presentation. There are some surprising omissions of key theorems in the subject. A noteworthy absentee is Frobenius’s theorem on the integrality of the central characters, a key result in proof of Burnside’s \(p^aq^b\) theorem (also omitted) and in Brauer’s development of block theory. Also missing is the proof of the existence of the kernel in a Frobenius group, a result still inaccessible without the use of group characters. The whole subject of Frobenius kernels and complements is one of such depth and applicability that it is a curious omission if one wishes to impress a newcomer by the power and variety of the method of group representations. Even if there is nothing new to provide in the way of proofs, the importance of the subject almost demands its inclusion. Another aspect of the book which we found unsatisfying is the lack of much chronology or attributions of proofs and concepts in the text. We could only find one reference to the date of a theorem, namely 1896, in connection with some work of Frobenius. The bibliography contains only three references to group representation theory, two of which are the books by Curtis and Reiner, and by Serre, and the other is the 1955 proof by Brauer and Tate of the Brauer induction theorem. The Schur relations are proved in Chapter 3 and used elsewhere, but no indication is given that these important formulae are due to Schur. We doubt if Theorem 3.8.2 is due to A. Speiser, since the theorem is universally acknowledged as first proved by Brauer and Nesbitt (1941). No reference is given for any proof by Speiser, and we have not found any mention of it in Speiser’s book on group theory. The Higman described as a discoverer of Theorem 7.1.16 is not uniquely determined by that name (D. G. Higman is intended) and the Kupich described among the second group of discoverers of the same theorem should be Kupisch.
The topics presented by the author include the orthogonality relations for irreducible complex characters, induced characters, Frobenius reciprocity, Mackey decomposition, Clifford’s theorem, Mackey’s method of little groups for constructing representations of the semi-direct product of an Abelian normal subgroup with some subgroup, introduction to modular representation theory. The Schur index is mentioned in passing and the Frobenius-Schur indicator is introduced and its properties developed, but none of its unusual applications is described. The high point of the text is probably the proof of the Brauer induction theorem using the Brauer-Tate method. This is then used to prove Schur’s conjecture that, for a finite group \(G\) of exponent \(n\), the field obtained by adjoining a primitive \(n\)-th root of unity to the rationals is a splitting field for \(G\). Given the power of the Brauer induction theorem, some other illustrations of its use might have been instructive. The author’s preference seems to be for ring-theoretic methods, and to some extent he develops the representation theory, especially in the modular case, using ideas from ring theory. He also pays some attention to the field of definition of a group representation, but, as we noted above, he does not develop this theme with the aid of the Schur index, which the Brauer induction theorem helps to study in some depth. We note that the discussion of integrality aspects of group representations, for example, the fact that the entries of representing matrices may be taken to be algebraic integers in some algebraic number field, relies on some unproved results in algebraic number theory. The usual device employed to avoid this higher theory is to make use of local integers, which suffice for many purposes.
To summarize, the author presents an unusual account of group representation theory. There are no exercises, few references, and no sense of the history, development and applications of the theory. On the positive side, the text reads well, the explanations are good and there are not many typographical errors. As the author is competing in a market place where high standards have been set, we feel that his offering displays surprising omissions and lacks those special features that will make its voice heard.

Reviewer: Roderick Gow (Dublin)

MSC:

20C15	Ordinary representations and characters
20C20	Modular representations and characters
20-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory

Keywords:

finite groups; group representations; group characters