Hook restriction coefficients
Sridhar P. Narayanan
TL;DR
This work tackles the restriction problem for GL_n representations after restricting to the S_n subgroup, with a focus on hook-shaped λ. It introduces a new nonrecursive expression for Specht polynomials q_μ and builds a generating-function framework to handle hook restriction coefficients r_{μ(k|ℓ)} via the Pieri rule, ultimately delivering a positive combinatorial interpretation in terms of supertableaux through two sign-reversing involutions. The key contributions are the nonrecursive Specht polynomial formula (Theorem th:specht), the generating-function formulation for hook coefficients, and the explicit combinatorial realization of r_{μ(k|ℓ)} and r_{μ[n](k|ℓ)} as fixed-point counts of SpT objects. These results illuminate the hook-case structure of restriction coefficients and connect to broader themes in character polynomials and FI-modules.
Abstract
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.