Dihedral Group Frames with the Haar Property

Abstract: We consider a unitary representation of the Dihedral group $$D_{2n}$$ obtained by inducing the trivial character from the co-normal subgroup $\left{0\right}\rtimes\mathbb{Z}_{2}.$ This representation is naturally realized as acting on the vector space $\mathbb{C}<sup>{n}.$ We prove that the orbit of almost every vector in $\mathbb{C}<sup>{n}$ with respect to the Lebesgue measure has the Haar property (every subset of cardinality $n$ of the orbit is a basis for $\mathbb{C}<sup>{n}$) if $n$ is an odd integer. Moreover, we provide explicit sufficient conditions for vectors in $\mathbb{C}<sup>{n}$ whose orbits have the Haar property. Finally, we derive that the orbit of almost every vector in $\mathbb{C}<sup>{n}$ under the action of the representation has the Haar property if and only if $n$ is odd. This completely settles a problem which was only partially answered in \cite{Oussa}.