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Representation theory of finite groups and associative algebras. Reprint of the 1962 original. (English) Zbl 1093.20003

Providence, RI: AMS Chelsea Publishing (ISBN 0-8218-4066-5/hbk). xiv, 689 p. (2006).
The book before us, usually referred to simply as Curtis and Reiner or more briefly, for the purpose of this review, as C and R, was first published in 1962. The bibliography was later extended to include references to papers that appeared up to 1966, and also to take account of some important omissions. This slightly revised edition was later republished as a (relatively cheap) paperback in 1988, and now the American Mathematical Society has made the book available again as a hardback reprint, reasonably priced at $80.
C and R is surely a classic text, which introduced many researchers to the subject of representation theory. The reviewer can testify to using the text to learn the rudiments of group representation theory, especially the modular theory, from 1970 onwards, and finding it very useful. Since the first appearance of C and R, other specialized texts have provided more detailed and helpful expositions of various aspects of representation theory. Thus, for example, a rather brief introduction to the subject of projective representations was presented in §51 and §53 of C and R, but B. Huppert’s Endliche Gruppen of 1967 gave a much more detailed explanation of the theory, based in part on a modern exposition of Schur’s original theory of the first decade of the twentieth century. Similarly, Huppert showed the power of what is now called Clifford theory, where a detailed analysis is made of the restriction to a normal subgroup of an irreducible group representation. A description of Clifford’s original 1937 theorem is given in §49 of C and R, and various applications of it are described, but one would not gain any feeling for its potential power in, say, the representation theory of finite solvable groups. Other books, such as W. Feit’s Characters of Finite Groups (1967) and I. M. Isaacs’s Character Theory of Finite Groups (1976) provide much more useful information about group characters and consequently, C and R is hardly the preferred port of call nowadays for this type of material. Of course, Curtis and Reiner published their own updated Methods of Representation Theory, which appeared in two volumes in 1981 and 1987. This text addressed some of the shortcomings of the original C and R, shortcomings which were mainly the result of new developments in the subject, but we suspect that it never acquired the dominant position in the research literature that the earlier work had achieved.
Opinions must surely vary, but we find that Chapter III, on algebraic number theory, Chapter IV on semi-simple rings and group algebras, Chapter VI on induced characters, Chapters VIII-X on non-semi-simple rings, Frobenius algebras, splitting fields and separable algebras, all remain valuable (this may reflect the reviewer’s lack of detailed knowledge on the ring theory aspects of the subject matter). Chapter XI, on integral representations, seems particularly useful to non-specialists who want a brief introduction to a notoriously demanding subject. For example, it is probably difficult to find descriptions of the classification of the integral representations of a group of prime order elsewhere. Overall, C and R may still be recommended to form part of the personal library of anyone interested in representation theory. Better sources now exist for some of its topics, but it is strong in a number of areas and is worth owning (or at least studying) for this reason. A fault, not uncommon in books of the time, is that not enough applications and examples are given. The use of only the dihedral group of order 12 and symmetric group \(S_4\) to illustrate the modular theory is scarcely illuminating, and no indication is given of the complexities of the subject, complexities which remain partly unresolved over 40 years later.
The book is remarkably free from typographical errors. We note the following amendments to the bibliography. The mathematician A. A. Bodi referred to twice is A. A. Bovdi, the correct pagination of the paper of Maschke is 492-498, and the name D. A. R. Wallace is missing, although three of his papers are listed, and one may also be omitted (this last is an accident of the updating of the references, since the name did appear in the 1962 version).

MSC:

20Cxx Representation theory of groups
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
16-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
01A75 Collected or selected works; reprintings or translations of classics
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20C10 Integral representations of finite groups
20C20 Modular representations and characters
16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
16L60 Quasi-Frobenius rings
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