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Publisher:

AMS/Chelsea

Publication Date:

2006

Number of Pages:

310

Format:

Hardcover

Edition:

2

Price:

45.00

ISBN:

0821840673

Category:

Textbook

[Reviewed by , on ]

Michael Berg

07/3/2006

Inside the back cover of this book, a 2006 Chelsea reprint, one finds the following description:

Originally written in 1940, this book remains a classic source on representations and characters of finite and compact groups. The book starts with necessary information about matrices, algebras, and groups. Then the author proceeds to representations of finite groups. Of particular interest in this part of the book are several chapters devoted to representations and characters of symmetric groups and the closely related theory of symmetric polynomials. The concluding chapters present the representation theory of classical Lie groups, including a detailed description of representations of the unitary and orthogonal groups. The book, which can be read with minimal prerequisites (an undergraduate algebra course), allows the reader to get a good understanding of beautiful classical results about group representations.

Indeed, this bit of propaganda is right on target; and it’s also revealing. For instance, since the book is almost seventy years old, the definition of “necessary information about matrices, algebras, and groups” is rather dated. To be sure, there is little or nothing present in the first three chapters (titled, respectively, “Matrices,” “Algebras,” “Groups”) which would be alien to a modern student with a good algebra background including e.g. Herstein’s *Topics in Algebra* (the best undergraduate mathematics book in creation, in this reviewer’s opinion) and a smidgen of stuff on algebras gleaned from more advanced texts. (Of course, Littlewood’s own treatment, at only ten pages, is elegant and readable. By the way, on p. 22 of the book, *Algebras and Their Arithmetics* , by Dickson, is referred, evincing the author’s superb taste.)

It should be stressed, however, that the reader would be ill-advised to come to the book under review without the background referred to above, given that Littlewood not only wastes no time, but arranges his presentation in what is today an alien style: the organization of the book is about as far away from, say, a Landau-type telegraph-style as can be imagined. One reads the book, pencil in hand, ready to supply details where needed — as with all good books, this should occur with great frequency.

As the above description suggests, the heart of the work is representation theory, the focus falling on finite groups (the symmetric group properly receives two chapters, the later devoted to its character theory) and compact Lie groups (the unitary group and its relatives, so to speak, rate a sixty-page chapter). Littlewood’s treatment of these topics is very heavily computational, or, more properly, computation-oriented. Working through this book will necessarily result in a variety of skills which are today probably regarded as somewhat anachronistic or, well, eccentric. But this modern attitude may well be wrongheaded, for the imposing calculations Littlewood carries out (pp. 228-232 are particularly rich in this regard) surround some very deep theorems. Anyone (algebraist, physicist, you name it) who would have a need to calculate a character table would be well-served by Littlewood’s book, as would anyone who learned representation theory from, for instance, Serre’s *Linear Representations of Finite Groups* (as holds true for me), but would like to see the nuts-and-bolts of the constructs exposed. Serre, as a representative *par excellence* of the modern style of doing this kind of mathematics, surely manifests a different focus. (For instance, the phrase, “induced representation,” doesn’t occur in Littlewood’s book; however, “induced matrix” does.)

In his Preface, Littlewood remarks that “[r]ather than to attempt any exhaustive treatment, it has been my aim to develop the nucleus of a theory which will bring to notice new problems to be solved.” So, factoring in the date of its writing (“all things new are old again”), *The Theory of Group Characters and Matrix Representations of Groups* counts as a fine source from which to learn a number of important algebraic techniques, beloved no doubt of the culture surrounding the classification problem, to pick an obvious example. It’s a beautifully written book, dated but by no means obsolete.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

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