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:
Supposeacts 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 isomorphismis canonical or functorial, where as the isomorphism
is not (it depends on a choice of basis).