Hook restriction coefficients

The permutation matrices form a subgroup of $\text{GL}_n(\mathbb{C})$ that is isomorphic to the symmetric group $S_n$. Let $r_{\mu\lambda}$ denote the multiplicity of the irreducible representation $V_\mu$ of $S_n$, corresponding to a partition $\mu$ of $n$, in the restriction of an irreducible polynomial representation $W_\lambda(\mathbb{C})$ of $\text{GL}_n(\mathbb{C})$, corresponding to a partition $\lambda$ with at most $n$ parts. Finding a combinatorial interpretation for $r_{\mu\lambda}$ remains an open problem in algebraic combinatorics, called the \emph{restriction problem}. We derive a new nonrecursive expression for a character polynomial called the \emph{Specht polynomial} and use it to find a combinatorial interpretation of $r_{\mu\lambda}$ when $\lambda$ is a hook-shaped partition.