Group Representation Theory, edited by Geck, Testermann, and Thévanez, constitutes something of an introductory course, and then some, on contemporary themes in group representation theory, stressing the categorical approach (at least to a certain degree), and on the more specialized subjects of algebraic groups and finite reductive groups. The book should be of great use to advanced graduate students as well as beginning researchers or workers in the field looking for new themes to investigate.
The book has two parts: the first presents five carefully crafted accounts, or expanded lecture note-sets, of such topics as block theory, fusion theory, and endo-presentation modules, as well as a good deal of cohomology; the second part of the book then deals specifically with algebraic groups and reductive groups. Thus the latter part aims toward a particular frontier, which fits well with the fact that the material in the book was first vetted in the setting of a research semester held in 2005 in Lausanne, Switzerland.
The pièce de résistance of part two, and of the entire book, is a three-part article by Jean-Pierre Serre, titled, “Bounds for the orders of finite subgroups of G(k),” coming at the very end of the book as a culmination of what came before. Prior to this, the reader is properly exposed to a good deal of the theory of algebraic groups, including the important subject of modular representations of Hecke algebras. I should also like to draw attention to the article by Seitz, immediately preceding the one by Serre, dealing, as it were, with the inner life of algebraic groups: this material stands very well on its own.
There is a lot of deep and interesting mathematics to be found in the pages of Group Representation Theory , and the chosen presentation makes for a quick ascent to these heights. This is a useful book, and looks to be a valuable addition to the library of any representation theorist with an interest in the themes covered here. This does not narrow the field much, given the surpassing importance of algebraic groups.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.