Stability of Restrictions of Representations of the Symmetric Group to the Hyperoctahedral Subgroup
Sergey Davydov
TL;DR
The paper establishes a stability phenomenon for restricting irreducible representations of the symmetric group $S_{2n}$ to the hyperoctahedral subgroup $H_n$, analogous to Murnaghan stability for Kronecker coefficients. It expresses the multiplicities via symmetric-function plethysm: the multiplicity of $V^{\lambda,\mu}$ in $\mathrm{Res}_{H_n}^{S_{2n}}(V^{\nu[2n]})$ corresponds to the coefficient of $s_{\nu}$ in $(s_{\lambda}\circ h_2)(s_{\mu}\circ e_2)$, with $n$-independent values for large $n$ as shown using Koike–Terada's framework. Two key lemmas are developed: (i) a stability lemma ensuring $K^{\nu}_{\lambda,\mu}(n)$ becomes constant when $2n \ge |\nu|+\nu_1+2|\lambda|+2|\mu|$, and (ii) a finiteness lemma proving that only finitely many pairs $(\lambda,\mu)$ contribute, via degree-based bounds and the function $\Phi$. Consequently, the decomposition $\mathrm{Res}_{H_n}^{S_{2n}}(V^{\nu[2n]}) = \bigoplus_{\lambda,\mu} \overline{K}_{\lambda,\mu}^{\nu} V^{\lambda[n-|\mu|],\ \mu}$ stabilizes, with a concrete bound $2n \ge 5|\nu|+\nu_1$, and a conjugate-stability statement holds under diagram transposition.
Abstract
The paper investigates the stability properties of restrictions of irreducible representations of the symmetric group to the hyperoctahedral subgroup. A stability result is obtained, analogous to the classical Murnaghan theorem on the stability of the decomposition of tensor products of representations of the symmetric group. The proof is based on the description of these restrictions in terms of symmetric functions from the K. Koike and I. Terada's paper.