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Representations of Groups: A Computational Approach

Cambridge University Press
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Representations of Groups: A Computational Approach, by Lux and Pahlings, is a well-constructed, dense, and, its introductory nature notwithstanding, pretty far-reaching text. Happily, it is also quite accessible.

Lux and Pahlings state in their Preface that their book addresses “representation theory of finite groups with some emphasis on the computational aspects of the subject.” There is in fact quite a lot of emphasis on computational aspects, seeing that, for instance, the book is overflowing with computer programs. Unfortunately computers are not my bailiwick and so I am of little use when it comes to evaluating the individual programs (which are easily picked out in the pages of the book: the font changes to something reminiscent of what an old typewriter would produce, but with an excellent ribbon). It is certainly the case, however, that the authors have gone to great pains to include a wealth of computational algorithms replete with machine opportunities.

Equally noteworthy, however, and more to my taste, is the fact that they have caught so many aspects of serious group representation theory in their nets. Chapter 1 covers “basics” and Chapter 2 covers group characters. Chapter 3 addresses “the interplay between representations of groups and subgroups,” with restriction (and fusion) dealt with in § 3.1, induction in § 3.2. But Lux and Pahlings also hit Clifford theory and projective representations (which I always think of as something of an underrepresented business, so I’m happy to see it here), and they finish with Brauer’s characterization of characters (“including some applications”).

This takes us to p. 288 (out of 441, omitting the section on references and the index), underscoring the fact that three long chapters are devoted to hard-core, but fundamental, group representation theory. The theory per se does not call for this much paper: for example, Serre’s classic Linear Representations of Finite Groups has only 170 pages, with (famously) a wealth of material on Brauer theory. But this is, again, testimony to the fact that Lux and Pahlings care greatly about the nuts and bolts of actual computation, and particularly the business of algorithms. To be sure, this is not only a distinguishing feature of Representations of Groups: A Computational Approach, it is a great virtue.

Speaking of Serre’s book, and specifically his vaunted exposition of Brauer theory, this pillar of modern group representation theory is certainly well represented also in the present book. Indeed Chapter 4 (at 441 – 288 = 153 pages), titled “Modular representations,” starts off with the following paragraph:

In the preceding chapters we studied representations of a group G over a fixed field K, where most of the time the characteristic of K was assumed not to be a divisor of |G|, so that the group algebra KG was semisimple. Often the study of representations of a finite group G over field of characteristic p dividing |G| is called “modular representation theory.” But this captures only a narrow aspect of the theory as it was developed by Richard Brauer. In fact, an important part of the theory is the interplay between representations of G in characteristic zero and in characteristic p.

After this innocuous prelude, they go on to cover a lot of serious Brauer theory, much of it done very explicitly (again, a great virtue).

Finally, as it should be given what the authors have in mind, Representations of Groups: A Computational Approach is laden with examples and exercises, many of them supplied with hints. The book is well-written and anything but chatty and is based on the authors’ courses so that much of the material in the book has been field tested — a good thing when so much computer programming is called for. The authors suggest using something called GAP (freely obtainable at a website they provide) for computer programming, but this just points to another gap in my knowledge in this area. So I had better quit while I’m ahead. Representations of Groups: A Computational Approach look good. It should do well.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Saturday, September 4, 2010
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Klaus Lux and Herbert Pahlings
Cambridge Studies in Advanced Mathematics 124
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Michael Berg
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Preface; Frequently used symbols; 1. Representations and modules; 2. Characters; 3. Groups and subgroups; 4. Modular representations; List of notation; Bibliography; Index.

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Tuesday, January 18, 2011