Two of the most interesting constructions involving group representations are restriction and induction. For our discussion of both of them, we let be a subgroup; it doesn’t have to be normal.
Now, given a representation , it’s easy to “restrict” it to just apply to elements of . In other words, we can compose the representing homomorphism with the inclusion : . We write this restricted representation as ; if we are focused on the representing space , we can write ; if we pick a basis for to get a matrix representation we can write . Sometimes, if the original group is clear from the context we omit it. For instance, we may write .
It should be clear that restriction is transitive. That is, if is a chain of subgroups, then the inclusion mapping is the exactly composition of the inclusion arrows and . And so we conclude that
So whether we restrict from directly to , or we stop restrict from to and from there to , we get the same representation in the end.
Induction is a somewhat more mysterious process. If is a left -module, we want to use it to construct a left -module, which we will write , or simply if the first group is clear from the context. To get this representation, we will take the tensor product over with the group algebra of .
To be more explicit, remember that the group algebra carries an action of on both the left and the right. We leave the left action alone, but we restrict the right action down to . So we have a -module , and we take the tensor product over with . We get the space ; in the process the tensor product over “eats up” the right action of on the and the left action of on . The extra left action of on leaves a residual left action on the tensor product, and this is the left action we seek.
Again, induction is transitive. If is a chain of subgroups, and if is a left -module, then
The key step here is that . But if we have any simple tensor , we can use the relation that lets us pull elements of across the tensor product. We get . That is, we can specify any tensor by an element in alone.