Noninjective induction of O G-interior algebras

Abstract

Let G and G’ be finite groups, φ: G → G’ a group homomorphism and A an O G-interior algebra. When φ is injective it is well-known how to construct the induced O G’-interior algebra Ind _φ , (A) (cf. 2.6) by mimicking the evident construction of End _O (Ind _φ (M)) from End _O (M) whenever M is an O-free O G-module. Although perhaps it is less evident, it turns out that such a mimicry is always possible, even when φ is not injective; we will apply this general construction in the following section to give an alternative description of the Hecke O G-interior algebras, which will appear naturally when dealing with Morita equivalences between Brauer blocks in Section 6 below.