Last update: 6 November 2012
In my work on the Brauer algebra, which is not a group algebra but is a semisimpie algebra with a distinguished basis, I have used the group algebra of the symmetric group as a guide, and tried to find generalizations to the Brauer algebra of as many of the properties of the symmetric group as possible. One of the outcomes of this work was the discovery that much of the representation theory of general semisimple algebras can be obtained in a fashion exactly analogous to the method used for finite groups. In this chapter I develop this theory from scratch. Along the way I prove, in the setting of semisimple algebras over an analogue of Maschke's theorem, a Fourier inversion formula, analogues of the orthogonality relations for characters and a formula giving the character of an induced representation, induced from a semisimple subalgebra. Section 4 reviews the double centralizer theory of I. Schur and defines a "Frobenius map" in the most general setting, a representation of a semisimple algebra. Such a map has proved useful in the study of the characters of the symmetric group, the Brauer algebra, and the Hecke algebra.
This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.