On the \(\mathrm{SO}(n+3)\) to \(\mathrm{SO}(n)\) branching multiplicity space (Q1632809)

Let \(n\geq2\). We embed the Lie group \(\mathrm{SO}(3)\) in the lower right hand corner of \(\mathrm{SO}(n + 3)\) and we embed \(\mathrm{SO}(n)\) in the upper left hand corner. First let \(n=2m\). Let the highest weight \(\mu\) of the irreducible representation \(\sigma\) of \(\mathrm{SO}(n)\) be given by the descending sequence \((\mu_1,\dots,\mu_m)\) and let the highest weight \(\lambda\) of the irreducible representation \(\pi\) of \(\mathrm{SO}(n+3)\) be given by the descending sequence \((\lambda_1,\dots,\lambda_{m+1})\). Here \(\mu_m\) and \(\lambda_{m+1}\) may be negative. Assume the sequences interlace, meaning that \(\lambda_i\geq |\mu_i|\geq\lambda_{i+1}\). Then the main result describes the branching multiplicity space \(\mathrm{Hom}_{\mathrm{SO}(n)}(\sigma,\pi)\) as an explicit tensor product of \(m+1\) representations of \(\mathrm{SO}(3)\). There is a similar result for the case \(n=2m+1\).