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

Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli

Abstract

We present a criterion for multiplicity-freeness of the decomposition of the restriction Res$^G_H(ρ_1 \otimes ρ_2)$ of the Kronecker product of two generic irreducible representations $ρ_1, ρ_2$ 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.

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

Abstract

We present a criterion for multiplicity-freeness of the decomposition of the restriction Res of the Kronecker product of two generic irreducible representations of a finite group with respect to a subgroup . This constitutes a generalization of a well known criterion due to Mackey (which corresponds to the case ). The corresponding harmonic analysis is illustated by detailed computations on the Clifford groups , together with the subgroups , for , which lead to an explicit decomposition of the restriction of Kronecker products.

Paper Structure

This paper contains 10 sections, 11 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Harmonic analysis for the pair $(G \times G \times H, \tilde{H})$
  4. A Frobenius reciprocity type theorem and proof of the main result
  5. Invariant vectors and spherical characters
  6. Harmonic analysis on Clifford groups
  7. Clifford groups and their irreducible representations
  8. Kronecker products of irreducible representations of ${\mathbb{CL}}(n)$
  9. The orbits of ${\mathbb{CL}}(n)$ on ${\mathbb{CL}}(n) \times {\mathbb{CL}}(n)$
  10. The spherical characters for $({\mathbb{CL}}(n) \times{\mathbb{CL}}(n) \times{\mathbb{CL}}(n), \widetilde{{\mathbb{CL}}(n)})$

Key Result

Theorem 1.1

Let $G$ be a finite group and $H \leq G$ a subgroup. Consider the subgroup $\widetilde{H}=\{(h,h,h):h\in H\}$ of $G\times G\times H$. Then the following conditions are equivalent:

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Definition 2.5
  • ...and 12 more