Introduction to orthogonal, symplectic and unitary representations of finite groups.
The book contains an introduction into the subjects of the title, but the subjects itself are subdivided into a lot of, mostly connected, topics. As such, inside Chapter 1, we find among others (semi)simple modules and algebras, involutions on algebras, formally real fields, bilinear and sesquilinear forms, Grothendieck groups, linear representations of finite groups. Likewise, inside Chapter 3 after the notion of isometry representations of finite groups (in fact a rather short Chapter 2), things on Hermitian forms over semisimple algebras are treated, such as Witt's theorem, the algebra of forms (containing Morita theory), the Hasse principle, and lifting a local representation to a global representation. In Chapter 4, equivariant Witt-Grothendieck and Witt groups are considered for several kinds of fields, as well as Schur subgroups, structure of Witt groups. Then, in Chapter 5, we find theory of representations over finite, local and global fields where symplectic, unitary and orthogonal representations are involved. In Chapter 6 (the last chapter of the book), we observe the theory of Fröhlich's invariant, Clifford algebras and the equivariant Brauer-Wall group. The book not only gives a wealth of information enabling the reader to do research for himself but also sidelines are mentioned with references in the list of literature. In conclusion, the book is an attractive addition to the existing literature on the subjects as considered.
scientific article; zbMATH DE number 5944361