S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 15: Restriction and Induction of Representations

15.1 Restriction

Throughout, is a finite group and a subgroup.

Definition 15.1.1 (Restriction). Given a representation of , the restriction of to , denoted , is viewed as a representation of (forget the action of elements outside ).

This defines a functor , where the categories have objects = finite-dimensional complex representations and morphisms = -equivariant (resp. -equivariant) linear maps.

An irreducible representation of need not remain irreducible upon restriction to . The problem we address next is: can we go in the opposite direction?