The induced representation

Charles Siegel over at Rigorous Trivialities suggested a NaNoWriMo for math bloggers: instead of writing a 50,000-word novel, just write a blog post every day. I have to admit I rather like the idea, so we’ll see if I can keep it up.

Continuing the previous post, what we want to do now is to think of restriction as a forgetful functor, since restricting a representation just corresponds to forgetting some of the data that defines it. Its left adjoint, if it exists, should be a construction of the “free -representation” associated to an -representation. Given a representation we therefore want to find a representation with the following universal property: any -intertwining operator for a -representation on naturally determines a unique -intertwining operator . In other words, we want to construct a functor such that

.

Some adjoint functors

Note: as usual, I will be playing free and loose with category theory in this post. Apologies to those who know better.

One way to define a subgroup of a group is as the image of a homomorphism into . Given the inclusion map , the functor in the category of groups acts contravariantly to give a map called restriction. More concretely, the restricted representation of a representation is defined simply by . Hence there is a functorial way to pass from a representation of a group to one of a subgroup .

It is not obvious, however, whether there is a functorial way to pass from a representation of back to one of . There is such a construction, which goes by the name of induction, and we will need it later. Today we’ll discuss the general category-theoretic context in which induction is understood, where it is called an adjoint functor. For more about adjoints, see (in no particular order) posts at Concrete Nonsense, the Unapologetic Mathematician, and Topological Musings.