Equivariant Map - Intertwiners

Intertwiners

A completely analogous definition holds for the case of linear representations of G. Specifically, if X and Y are the representation spaces of two linear representations of G then a linear map f : X → Y is called an intertwiner of the representations if it commutes with the action of G. Thus an intertwiner is an equivariant map in the special case of two linear representations/actions.

Alternatively, an intertwiner for representations of G over a field K is the same thing as a module homomorphism of K-modules, where K is the group ring of G.

Under some conditions, if X and Y are both irreducible representations, then an intertwiner (other than the zero map) only exists if the two representations are equivalent (that is, are isomorphic as modules). That intertwiner is then unique up to a multiplicative factor (a non-zero scalar from K). These properties hold when the image of K is a simple algebra, with centre K (by what is called Schur's Lemma: see simple module). As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.