Exercises from Week 1

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 $\mathbb{C}$ (e.g. vector spaces over $\mathbb{F}_2$ ). What can go wrong?
If you choose two different characters (degree 1 representations) of a group, why aren’t they isomorphic?
Let $G$ be a finite group. How many different degree 1 representations does $G$ have?
Consider a group $G$ which acts on a set $X$ , and show that this induces a corresponding action on $Fun(X)$ (the set of all functions mapping from $X$ into some other set, say $\mathbb{C}$ ). Show that the regular representation of $G$ is a particular case of this.
Theorem 1 of Serre states: Let $\rho: G\to GL(V)$ be a representation, and let $W$ be a subspace of $V$ which is stable under $\rho$ ; then there is a complement space $W^c$ which is also stable.
Write a proof of this theorem in your own words. Highlight where we use the following facts: (1) $G$ is finite; (2) $V$ is finite-dimensional; (3) $V$ is over the field $\mathbb{C}$ .
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 $G$ is a group, why is $F(G)$ (the free vector space on $G$ ) the same as the regular representation of $G$ ?
For vector spaces $U,V,W,X$ , does it hold that $\textrm{Hom}(U,W)\otimes\textrm{Hom}(V,X) \simeq \textrm{Hom}(U\otimes V,W\otimes X)$ ?
If $U$ and $V$ are finite dimensional vector spaces, prove that $(U \otimes V)^* = U^* \otimes V^*$ . Show that this does not hold in general when $U, V$ are infinite-dimensional.
For vector spaces $U, V, W$ , prove that $\textrm{Hom}(U\otimes V,W)=\textrm{Hom}(U,\textrm{Hom}(V,W))$ .
Let $V$ be a finite dimensional vector space, and $V^*$ denote its dual and $V^{**}=(V^*)^*$ denote its double dual. One knows that $V$ is isomorphic to $V^*$ and to $V^{**}$ . However, the isomorphism $V\simeq V^{**}$ is much better than the isomorphism $V\simeq V^*$ !! Here is one reason:

Suppose $G$ acts on $V$ ; i.e., we have a representation $\rho:G\to GL(V)$ . Show that $G$ acts on $V^*$ and $V^{**}$ . Let us denote these representations by $\rho^*$ and $\rho^{**}$ . Show that the representations $(\rho, V)$ and $(\rho^{**}, V^{**})$ are always isomorphic, where as, it could happen that $(\rho,V)$ and $(\rho, V^*)$ are not isomorphic!!
One says that the isomorphism $V \simeq V^{**}$ is canonical or functorial, where as the isomorphism $V\simeq V^*$ is not (it depends on a choice of basis).