REPRESENTATION THEORY OF FINITE GROUPS

PROBLEMS SET 7

Date: 19th June 2017.

  1. Perform the combinatorial resolution of the partition representations of S4 to contrcut all the irreducible representations of S4. Use this to derive the character table of S4 from first principles.
  2. Let G be a finite group, χ : G → C* be a character of G. Let X and Y be G-sets. Given x ∈ X and y ∈ Y , show that there exists a function k : X × Y → C with k(x,y)≠0 if and only if, for every g ∈ G such that g ⋅ x = x and g ⋅ y = y, χ(g) = 1.
  3. Let λ be a partition of n, and (ρ,C[Xλ]) be the corresponding partition representation. Show that det(ρ(w))) = 1 for all w ∈ Sn if and only if
    ( ) ∑ n - 2 λ1,...,λi-1,λi - 1,λi+1,...,λj-1,λj - 1,λj+1, ...,λl 1≤i<j≤l

    is even (this is a multinomial coefficient of n - 2. In the lower row, the ith and jth parts of λ are reduced by 1).

  4. Show that [Sn,Sn] = An.