REPRESENTATION THEORY OF FINITE GROUPS

PROBLEMS SET 9

Date: 21th June 2017.

  1. Show that An is generated by s1si, with 2 ≤ i ≤ n - 1. Here, as usual si denotes the simple transposition (i,i + 1).
  2. Given a representation ρ : G → GL(V ) of a group, and an automorphism σ : G → G, let ρσ denote the representation ρ ∘ σ : G → GL(V ).
    1. What is the relationship between the character of ρ and the character of ρσ?
    2. Show that, if σ is an inner automorphism, then ρσ and ρ are isomorphic as a representations of G.
    3. If σ and τ are two automorphisms of G, and τ-1σ is an inner automorphism of G, then ρσ and ρτ are isomorphic as representations of G.
  3. For w ∈ An, show that the conjugacy class of w in Sn is a union of two classes in An if and only if w has distinct odd parts.
  4. In the character table of A4 compute χ(2,2)± at the three-cycles (1, 2, 3) and (2, 1, 3).
  5. In the character table of A5, compute χ(3,1,1)± at the 5-cycles (1, 2, 3, 4, 5) and (2, 1, 3, 4, 5).
  6. Enumerate all the conjugacy classes and irreducible representations of A8.