Hook restriction coefficients

Abstract

The permutation matrices form a subgroup of ${\text{GL}}_n(\mathbb{C})$ that is isomorphic to the symmetric group $S_n$. Let $r_{\hat{I}¼\hat{I}»}$ denote the multiplicity of the irreducible representation $V_{\hat{I}}¼$ of $S_n$, corresponding to a partition $\hat{I}¼$ of $n$, in the restriction of an irreducible polynomial representation $W_{\hat{I}}»(\mathbb{C})$ of ${\text{GL}}_n(\mathbb{C})$, corresponding to a partition $\hat{I}»$ with at most $n$ parts. Finding a combinatorial interpretation for $r_{\hat{I}¼\hat{I}»}$ 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 $r_{\hat{I}¼\hat{I}»}$ when $\hat{I}»$ is a hook-shaped partition.

12 pages, 5 figures