Representations and cohomology. I: Basic representation theory of finite groups and associative algebras. (English) Zbl 0718.20001

Benson, D. J.

Cambridge Studies in Advanced Mathematics, 30. Cambridge etc.: Cambridge University Press. xi, 224 p. £25.00; $ 39.50 (1991).

The book under review is the first of two volumes dedicated to representation theory, cohomology theory, and their interplay. The wealth of material presented in little more than 200 pages is impressive, and the treatment is elegant throughout.
The book is divided into six chapters. The first three of these are introductory in nature and present background material from rings and modules, homological algebra and modules for group algebras. Among more classical topics (like vertices, sources, Green correspondence and Green’s indecomposability theorem) there are also a short course on Morita theory, an outline of how concepts for group algebras generalize to Hopf algebras, theorems by Green and Maranda on the liftability of modules and Quillen’s approach to Jennings’ theory of p-group algebras.
The fourth chapter gives an introduction into Auslander-Reiten theory of finite-dimensional algebras. It starts with hereditary algebras, path algebras of quivers and the quiver of an algebra. The indecomposable representations of the Klein four group are constructed. There is a discussion on the representation type of quivers and algebras containing Gabriel’s theorem. Functor categories and the Auslander algebra are introduced. The author presents the method of functorial filtrations which is then applied to the construction of the indecomposable representations of the dihedral groups. Next we get to Auslander-Reiten sequences (for finite-dimensional algebras), irreducible morphisms and Auslander’s proof of Rojter’s theorem. The chapter concludes with the results by Riedtmann and Webb on the structure of Auslander-Reiten quivers, and with a discussion of Brauer graph algebras.
The fifth chapter on representation and Burnside rings starts with Grothendieck rings, both in characteristic 0 and prime characteristic. The Burnside and trivial source rings are applied to give a new approach to the induction theorems of Artin, Brauer, Conlon and Dress. The author continues with a construction of various ideals in representation rings and discusses a quotient without nilpotent elements and the semisimplicity question. He presents the Brauer lift and two proofs of the fact that the determinant of the Cartan matrix is a power of p. The chapter concludes with the Benson-Parker results on bilinear forms on representation rings.
The final chapter of the book is devoted to block theory (mainly in prime characteristic). It contains the definition of defect groups, Brauer’s three main theorems (the second one in Nagao’s form) and the structure of blocks of defect zero and of blocks with central defect groups. Its highlights are the treatment of the theory of blocks with cyclic or Klein four defect groups.
It can be seen from this description of its contents that the present volume has some overlap with the author’s previous book. It is a welcome update and much broader in perspective. Large parts are well suited for lectures on the subject.

Reviewer: B.Külshammer (Augsburg)

MSC:

20-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20C20	Modular representations and characters
20C05	Group rings of finite groups and their modules (group-theoretic aspects)
16-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
16G70	Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
20-02	Research exposition (monographs, survey articles) pertaining to group theory
19A22	Frobenius induction, Burnside and representation rings
20J05	Homological methods in group theory
20J06	Cohomology of groups
16S34	Group rings
20C11	\(p\)-adic representations of finite groups
16Gxx	Representation theory of associative rings and algebras

Keywords:

group algebras; vertices; sources; Green correspondence; finite- dimensional algebras; hereditary algebras; path algebras; quivers; indecomposable representations; representation type; Auslander-Reiten sequences; Auslander-Reiten quivers; Burnside rings; Grothendieck rings; induction theorems; representation rings; Cartan matrix; defect groups; Brauer’s three main theorems; blocks

Citations:

Zbl 0564.20004