Here’s a list of some questions which had been left unanswered from the week 1 meeting:
- Consider representations on vector spaces where the base field is not (e.g. vector spaces over ). What can go wrong?
- If you choose two different characters (degree 1 representations) of a group, why aren’t they isomorphic?
- Let be a finite group. How many different degree 1 representations does have?
- Consider a group which acts on a set , and show that this induces a corresponding action on (the set of all functions mapping from into some other set, say ). Show that the regular representation of is a particular case of this.
- Theorem 1 of Serre states: Let be a representation, and let be a subspace of which is stable under ; then there is a complement space which is also stable.
Write a proof of this theorem in your own words. Highlight where we use the following facts: (1) is finite; (2) is finite-dimensional; (3) is over the field .
For each of the three conditions, find a counter-example to show that the theorem doesn’t hold if the condition is dropped.
(This will also be a midterm exam question.)
- If is a group, why is (the free vector space on ) the same as the regular representation of ?
- For vector spaces , does it hold that ?
- If and are finite dimensional vector spaces, prove that . Show that this does not hold in general when are infinite-dimensional.
- For vector spaces , prove that .
Let be a finite dimensional vector space, and denote its dual and denote its double dual. One knows that is isomorphic to and to . However, the isomorphism is much better than the isomorphism !! Here is one reason:Suppose acts on ; i.e., we have a representation . Show that acts on and . Let us denote these representations by and . Show that the representations and are always isomorphic, where as, it could happen that and are not isomorphic!!
One says that the isomorphism is canonical or functorial, where as the isomorphism is not (it depends on a choice of basis).