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 ResG H(1 ⊗ 2) of the Kronecker product of two generic irreducible represen- tations 1, 2 of a finite group G with respect to a subgroup H ≤ 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 = CL(n), together with the subgroups H = CL(n − 1), for n ≥ 1, which lead to an explicit decomposition of the restriction of Kronecker products.