REPRESENTATION THEORY OF FINITE GROUPS

PROBLEMS SET 10

Date: 22nd June 2017.

  1. For a permutation w ∈ Sn define:
    ∏ ---i---j---- σ(w ) = w (i) - w(j) . 1≤i<j≤n

    Show that σ(w) coincides with the sign character value ϵ(w).

  2. Show that the alternating group An is generated by the 3-cycles (1, 2, 3), (1, 2, 4),…, (1, 2,n).
  3. If n ≥ 5, show that there is no nontrivial conjugacy class in An with fewer than n elements.
  4. Given the character values of S9:
    χ(5,14)(w(9)) = 1,
    χ(5,14)(w(5,3,1)) = 0,
    χ(33)(w(9)) = 0, and
    χ(33)(w(5,3,1)) = -1,
    compute the character values of A9:
    χ± (w ± ) and χ± (w ± ). (5,14) (9) (33) (5,3,1)