Both linear representations of finite groups and the theory of quadratic, skew-symmetric and Hermitian forms are classical subjects, with a wealth of results and interactions with larger sectors of mathematics. Hence, the idea that these two large traditional subjects should be related in a fundamental way is one of those ideas whose time was overdue.
As the author of the book under review reminds us, it was A. Fröhlich who first gave a systematic organization of the deep inter-relations between these two areas. In a series of papers in the 1960s, he studied representations of (finite) groups on vector spaces with a symmetric or skew-symmetric bilinear form. The main objective of the theory is to find invariants of finite groups using representations on orthogonal, unitary or symplectic vector spaces. For the classical fields (local, global, finite, algebraically closed or real closed fields), several specific invariants for finite groups are obtained, bringing into play the classical theory of quadratic, symmetric and Hermitian forms. Most of these invariants are obtained from the Grothendieck-Witt ring of orthogonal, unitary or symplectic representations or the Clifford algebra of an orthogonal representation, with the classical Brauer group of the base field replaced by the equivariant Brauer-Wall group.
The book under review provides an introduction to these developments assuming a basic background on linear representations of finite groups, as in Serre’s Linear Representations of Finite Groups (Springer, 1977), and the classical theory of quadratic, symplectic and Hermitian forms, as in Lam’s Introduction to Quadratic Forms over Fields (AMS, 2004), and involutions over simple algebras, as in Knus, Merkurjev, Rost and Tignol’s The Book of Involutions (AMS, 1998). Most of these results are collected, without proofs, in the first chapter. Chapter two gives a fast-paced introduction to isometry representations, i.e., representations on a vector space with a symmetric, skew-symmetric, Hermitian or skew-Hermitian bilinear form and which are invariant under the corresponding form. In the case of linear representations, many of the properties of the representations are translated into the language of modules over a group algebra; here this role is played by the Hermitian forms over a semisimple algebra, and all the necessary results are collected in chapter three.
Chapter four recalls all needed results from the Witt and Grothendieck-Witt groups of isometry representations and their functorial properties. Several examples are examined in detail: algebraically closed, real closed, and formally real fields. This chapter ends with a determination of the torsion subgroup of the Witt group of a simple K-algebra with involution, a result of W. Scharlau. Chapter five treats in detail the isometry representations over finite, local and global fields, proving the main (partial) results obtained so far. This chapter ends with a section dealing with isometry representations of the symmetric group, where as for the case of linear representations, the isometry representations of the symmetric group are rather explicit. The final chapter six, following C. T. C. Wall, introduces the equivariant Brauer-Wall group that gives several important invariants for orthogonal representations.
Riehm′s book is a most welcome introduction to this wonderful subject with deep roots in two classical areas of mathematics, collecting in one place the most recent developments. It can be profitably read by anyone interested with a basic background on representation theory and quadratic forms.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.