  Finding a combinatorial interpretation of the multiplicities of each
irreducible representation of the symmetric group $S_n$ in the restriction of
an irreducible polynomial representation of the general linear group
$GL_n(\mathbb{C})$ remains an interesting open problem in algebraic
combinatorics. In this paper we interpret the multiplicities of all irreducible
representations of $S_n$ in the restriction of an irreducible polynomial
representation of $GL_n(\mathbb{C})$ indexed by a hook partition.
