Handbook of Linear Algebra, 2nd Edition

Chapter 85

Group Representations

Randall R. Holmes

Auburn University

Tin-Yau Tam

Auburn University

Representation theory is the study of the various ways a given group can be mapped into a general linear group. This information has proven to be effective at providing insight into the structure of the given group as well as the objects on which the group acts. Most notable is the central contribution made by representation theory to the complete classification of finite simple groups [Gor94]. (See also Fact 85.1.3 and Fact 85.6.5.)

Representations of finite groups can be defined over an arbitrary field and such have been studied extensively. Here, however, we discuss only the most widely used, classical theory of representations over the field ...