On the SO(n+3) to SO(n) branching multiplicity space

TL;DR

This paper analyzes how the multiplicity space for the branching from SO(n+3) to SO(n) decomposes as an SO(3)-module, revealing a tensor product structure under certain interlacing conditions of highest weights.

Contribution

It provides a detailed decomposition of the SO(n+3) to SO(n) multiplicity space as an SO(3)-module, highlighting a tensor product structure when highest weights interlace.

Findings

Decomposition as tensor product of SO(3) representations

Interlacing of highest weights determines structure

Explicit description of multiplicity space structure

Abstract

We study the decomposition as an $\textrm{SO}(3)$-module of the multiplicity space corresponding to the branching from $\textrm{SO}(n+3)$ to $\textrm{SO}(n)$. Here, $\textrm{SO}(n)$ (resp.\ $\textrm{SO}(3)$) is considered embedded in $\textrm{SO}(n+3)$ in the upper left-hand block (resp.\ lower right-hand block). We show that when the highest weight of the irreducible representation of $\textrm{SO}(n)$ interlaces the highest weight of the irreducible representation of $\textrm{SO}(n+3)$, then the multiplicity space decomposes as a tensor product of $\lfloor (n+2)/2\rfloor$ reducible representations of $\textrm{SO}(3)$.