92m:20005 20Cxx 16-02 16Gxx 20-02 20J06

Benson, D. J. [Benson, David John] ^{(}4-OX^{)}

⋆Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Cambridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge, 1991. xii+224 pp. $39.50. ISBN 0-521-36134-6 This first volume is a very welcome addition to the existing classical literature on representation theory and cohomology of finite groups [87i:20002J. L. Alperin, Local representation theory, Cambridge Univ. Press, Cambridge, 1986; MR 87i:20002; C. W. Curtis and I. Reiner, Methods of representation theory. Vol. I, Wiley, New York, 1981; MR 82i:20001; ibid., Vol II, 1987; MR 88f:20002; W. Feit, The representation theory of finite groups, North-Holland, Amsterdam, 1982; MR 83g:20001; K. S. Brown, Cohomology of groups, Springer, New York, 1982; MR 83k:20002; S. Lang, Rapport sur la cohomologie des groupes, Benjamin, New York, 1967; MR 35#2948].

This book cannot be compared to any of them—not only because of the author’s individual choice of topics, but also because he sets up his own standards.

Let me briefly discuss the contents. After a short introduction to rings and modules, homology and cohomology (with Yoneda composition) is introduced, including induction and restriction. Kunneth’s theorem is proved (the statement on p. 39, Theorem 2.7.1 “short exact sequence of Λ-modules”, is not true without further assumptions); Ext and Tor for chain complexes and hyper(co)homology are discussed.

In the next chapter on modules for group algebras, standard representation theory—p-adic and modular—is developed: Tensor- and cup-products are considered for cocommutative Hopf algebras (e.g., the Benson-Carlson criterion of when a tensor product splits off from a trivial module is proved for a Hopf algebra with an antipode). On the cohomology side, induction (including tensor induction), restriction and transfer, as well as relative cohomology, are discussed and the group-theoretic interpretation of low-dimensional cohomology is used to give a very elegant proof of Maranda’s and Green’s lifting theorems. On the side of representation theory, the Green correspondence and results from Clifford theory are proved. Concerning Clifford theory, the author writes: “We present a primitive version of Clifford theory”; he discusses many topics in great depth, and I think that more details on Clifford theory—say as in Curtis and Reiner’s book— would have been desirable. This chapter concludes with Jennings’ and Quillen’s description of the p-restricted Lie algebra, associated to a p-group.

The next chapter is a crash course in modern representation theory of Artin algebras: quivers (with relation) and their representations, hereditary algebras as path algebras, a finite-dimensional algebra as quotient of its Ext-quiver, the description of the indecomposable representations of Klein’s four-group in characteristic two, the trichotomy—finite, tame (domestic) and wild—is discussed (group rings are classified accordingly). Here most results are quoted without proofs. The next section, on Dynkin and Euclidean diagrams and the representation type, Coxeter transformations, Weyl groups, the Auslander algebra (not only for finite representation type), almost split sequences and the Auslander-Reiten quiver, contains proofs. The results of Riedtmann, Happel-Ringel-Preiser and Webb on the structure of the stable Auslander-Reiten quiver are derived and applied to construct the various components of the Auslander-Reiten quiver of Klein’s four-group. In preparation for the structure of blocks with cyclic defect, general Brauer graph algebras are discussed. This chapter deals only with Artinian algebras, and corresponding results for p-adic representations are not mentioned.

Chapter 5 discusses representation rings and Burnside rings: Dress’s congruences for the Burnside ring are derived, and tom Dieck’s proof of Dress’s characterisation of solvable groups in terms of idempotents in the Burnside ring is presented. Next comes the result of Benson-Parker, that the complex form of the representation ring is the direct sum of the image of induction and the kernel of restriction. The classical induction theorems for representation- and Grothendieckrings by Artin, Brauer, Conlon and Dress are presented, and a large semi-simple quotient of the representation ring is constructed (Benson-Carlson) extending the results of Green and O’Reilly on nilpotent elements. Finally—using almost split sequences—two nondegenerate bilinear forms are constructed (Benson-Parker).

The final chapter consists of standard block theory: the three main theorems of Brauer (extended version), Clifford theory for blocks, and a description of blocks with cyclic defect (Brauer, Thompson, Green and Dade) using the previously derived results on Brauer algebras. The Auslander-Reiten quiver of these blocks is described. The final section is devoted to a sketch of Erdman’s description of blocks with Klein’s four-group as defect group. In this section the p-adic block thory is reduced to the modular one; if the reader is not very careful, he might get a confused intuition for the p-adic situation.

The bibliography contains a carefully selected list of 210 titles. The sections contain exercises, which in most cases push the theory further on and are by no means easy. There are numerous comments and remarks, which point out connections, explain and summarize results and give historical developments. It is surprising how much material can be presented clearly in 210 pages.

In his approach to representation theory of finite groups, the author takes entirely and consistently the module-theoretic point of view, e.g., character theory is the study of homomorphisms from the complex representation ring to the complex numbers.

This book is fascinating reading for someone who already has a basic knowledge of representation theory and cohomology, since only he can appreciate the proofs, which are very elegant and often new. These proofs are stripped of anything superfluous, stress the essential ideas and demonstrate the author’s deep insight and masterly knowledge of the subject. This also makes it sometimes hard to follow the arguments. To quote the author: “The pace is brisk.”

I can recommend this book highly to anyone who works with representation theory and cohomology of finite groups.

K. W. Roggenkamp (D-STGT-B)

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