The group chain S On,1⊃S O1,1⊗S On−1, a complete solution to the ``missing label'' problem

We discuss the decomposition $\mathbf{S}\hspace{0.17em}{\mathbf{O}}_{\text{n,1}}\supset \mathbf{S}\hspace{0.17em}{\mathbf{O}}_{\text{1,1}}\otimes \mathbf{S}\hspace{0.17em}{\mathbf{O}}_{\text{n−1}}$ by constructing multiplier representations over the group manifold of S O _n. Explicit orthogonal and complete bases in terms of functions diagonal with respect to the canonical $(\mathbf{S}\hspace{0.17em}{\mathbf{O}}_{\text{n,1}}\supset \mathbf{S}\hspace{0.17em}{\mathbf{O}}_n)$ and noncanonical $(\mathbf{S}\hspace{0.17em}{\mathbf{O}}_{\text{n,1}}\supset \mathbf{S}\hspace{0.17em}{\mathbf{O}}_{\text{1,1}}\otimes \mathbf{S}\hspace{0.17em}{\mathbf{O}}_{\text{n−1}})$ chains are provided which give a complete solution to the ``missing label'' and multiplicity problems occuring in the latter decomposition. Moreover, an integral representation for the overlap functions between the two chains is given, for which the singularity structure can be immediately ascertained. Expressions for the cases n = 3 and 4 are given.