I am struggling with these exercise from group representations and would really appreciate some steps to take or sources with similar exercises.
The task is to compute the character of the irreducible representation $ψ^λ$ : $S_4$ → $Aut_C(S^λ)$ for each λ $\vdash 4$.
And then verify that $ψ^λ$ when expressed in the matrix form with respect to the standard basis is defined over $\mathbb{Z}$ (matrices have integral coefficients).
DEFINITION: A partition of $n$ (denoted λ $\vdash n$) (is a tuple $λ = (λ_1, λ_2, . . . , λ_l)$ is a tuple for which $λ_1 ≥ λ_2 ≥ · · · ≥ λ_l ≥ 1$ and $\sum_{i=1}^{l} \lambda_i = 1$.
This fact may also be useful: if n ∈ N, λ $\vdash$ n, then the Specht module $S^λ$ has a basis consisting exactly of all polytabloids associated to standard tableaux.