Exercises from Week 1

Here’s a list of some questions which had been left unanswered from the week 1 meeting:

  1. 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?
  2. If you choose two different characters (degree 1 representations) of a group, why aren’t they isomorphic?
  3. Let G be a finite group. How many different degree 1 representations does G have?
  4. 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.
  5. 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.)
  6. If G is a group, why is F(G) (the free vector space on G) the same as the regular representation of G?
  7. 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)?
  8. 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.
  9. For vector spaces U, V, W, prove that \textrm{Hom}(U\otimes V,W)=\textrm{Hom}(U,\textrm{Hom}(V,W)).
  10. 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).

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