## Tohoku Mathematical Journal 2015 December SECOND SERIES VOL. 67, NO. 4

 Tohoku Math. J. 67 (2015), 553-571

Title MACKEY'S CRITERION FOR SUBGROUP RESTRICTION OF KRONECKER PRODUCTS AND HARMONIC ANALYSIS ON CLIFFORD GROUPS

Author Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli

(Received January 28, 2014, revised July 30, 2014)
Abstract. We present a criterion for multiplicity-freeness of the decomposition of the restriction ${\rm Res}^G_H(\rho_1 \otimes \rho_2)$ 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.

Mathematics Subject Classification. Primary 20C15; Secondary 43A90, 20G40.

Key words and phrases. Representation theory of finite groups, Gelfand pair, Mackey's criterion, Kronecker product, Clifford groups.