In this chapter we discuss the notion of an induced representation and the structure of the commutant of a representation, and we present a new approach to Clifford theory. We assume the reader to be familiar with the basic rudiments of the representation theory of finite groups. We refer to the monographs by Bump [7], Fulton and Harris [29], Isaacs [36], Serre [67], Simon [68] and Sternberg [73] as basic references; see also our monograph [15].

In Section 1.1 we present the main properties of induction, focusing on the Frobenius character formula and Frobenius reciprocity. Then, in Section 1.2, we discuss several aspects of Frobenius reciprocity for a permutation representation; in particular, we show that the spherical Fourier transform provides an explicit isomorphism between the commutant of a permutation representation and the algebra of bi-K-invariant functions. In the last part of the section we examine the particular case of a multiplicity free permutation representation, which yields the notion of a Gelfand pair. Finally, in Section 1.3, we present an exposition of Clifford theory, which provides a powerful tool for relating the representation theory of a group G and the representation theory of a normal subgroup N ≤ G.

Induced representations

The presentation of this section was inspired by the books by Bump [7], Serre [67] and Sternberg [73] and by our research-expository paper [12].

1.1.1 Definitions

Let G be a finite group. Let K be a subgroup of G and (ρ, W) a representation of K.