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$^G_H(\rho_1 \otimes \rho_2)$ of the Kronecker product of two generic irreducible representations $\rho_1, \rho_2$ of a finite group $G$ with respect to a subgroup $H \leq G$. 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 $G={\mathbb{CL}}(n)$, together with the subgroups $H={\mathbb{CL}}(n-1)$, for $n \geq 1$, which lead to an explicit decomposition of the restriction of Kronecker products.