Some Restriction Coefficients for the Trivial and Sign Representations

We use character polynomials to obtain a positive combinatorial interpretation of the multiplicity of the sign representation in irreducible polynomial representations of $G{L}_n(\mathbb{C})$ indexed by two-column and hook partitions. Our method also yields a positive combinatorial interpretation for the multiplicity of the trivial representation of $S_n$ in an irreducible polynomial representation indexed by a hook partition.