# Dihedral Group Frames with the Haar Property

## TL;DR

This paper proves that for odd integers n, the orbit of almost every vector under a specific dihedral group representation forms a basis for ^n, establishing the Haar property and providing explicit conditions.

## Contribution

It demonstrates that for odd n, almost all vectors have orbits with the Haar property under the dihedral group representation, fully resolving a previously partial problem.

## Key findings

- Almost every vector's orbit has the Haar property for odd n.
- Explicit conditions are given for vectors with the Haar property.
- The Haar property holds if and only if n is odd.

## Abstract

We consider a unitary representation of the Dihedral group $D_{2n}% =\mathbb{Z}_{n}\rtimes\mathbb{Z}_{2}$ 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 $\mathbb{C}^{n}.$ We prove that the orbit of almost every vector in $\mathbb{C}^{n}$ with respect to the Lebesgue measure has the Haar property (every subset of cardinality $n$ of the orbit is a basis for $\mathbb{C}^{n}$) if $n$ is an odd integer. Moreover, we provide explicit sufficient conditions for vectors in $\mathbb{C}^{n}$ whose orbits have the Haar property. Finally, we derive that the orbit of almost every vector in $\mathbb{C}^{n}$ 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}.