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 $r_{μλ}$ denote the multiplicity of the irreducible representation $V_μ$ of $S_n$, corresponding to a partition $μ$ of $n$, in the restriction of an irreducible polynomial representation $W_λ(\mathbb{C})$ of $\text{GL}_n(\mathbb{C})$, corresponding to a partition $λ$ with at most $n$ parts. Finding a combinatorial interpretation for $r_{μλ}$ 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 $r_{μλ}$ when $λ$ is a hook-shaped partition.