Mackey's criterion for subgroup restriction of Kronecker products and harmonic analysis on Clifford groups

We present a criterion for multiplicity-freeness of the decomposition of the restriction Res${}_H^G({\rho }_1\otimes {\rho }_2)$ of the Kronecker product of two generic irreducible representations ${\rho }_1,{\rho }_2$ of a finite group $G$ with respect to a subgroup $H\le G$. This constitutes a generalization of a well known criterion due to Mackey (which corresponds to the case $H=G$). The corresponding harmonic analysis is illustated by detailed computations on the Clifford groups $G=\mathbb{C}\mathbb{L}(n)$, together with the subgroups $H=\mathbb{C}\mathbb{L}(n-1)$, for $n\ge 1$, which lead to an explicit decomposition of the restriction of Kronecker products.