The Classical Groups: Their Invariants and Representations

In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. Analysis and topology are used wherever appropriate. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics.

In The Classical Groups, his most important book, Weyl provided a detailed introduction to the development of group theory, and he did it in a way that motivated and entertained his readers. Departing from most theoretical mathematics books of the time, he introduced historical events and people as well as theorems and proofs. One learned not only about the theory of invariants but also when and where they were originated, and by whom. He once said of his writing, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful."

Contents

1. Introduction
2. Vector Invariants
3. Matric Algebras and Group Rings
4. The Symmetric Group and the Full Linear Group
5. The Orthogonal Group
6. The Symplectic Group
7. Characters
8. General Theory of Invariants
9. Matric Algebras Resumed
Supplements
Errata and Addenda
Bibliography
Supplementary Bibliography, Mainly for the Year 1940-1945
Index