Induced representations; Mackey’s criterion

Abstract

Let H be a subgroup of a group G and R a system of left coset representatives for H. Let V be a C[G]-module and let W be a sub-C[H]module of V. Recall (cf. 3.3) that the module V (or the representation V) is said to be induced by W if we have V = ⊕_s∈RsW, i.e., if V is a direct sum of the images sW, s E R (a condition which is independent of the choice of R). This property can be reformulated in the following way: Let

$$ W' = C\left[ G \right]{ \otimes _{C\left[ H \right]}}W $$

be the C[G]-module obtained from W by scalar extension from C[H] to C[G]. The injection W → V extends by linearity to a C[G]-homomorphism i: W′→V.