Dihedral Group Frames which are Maximally Robust to Erasures

Abstract

Let $n$ be a natural number larger than two. Let $D_{2n}=\langle r,s : r^{n}=s^{2}=e, srs=r^{n-1} \rangle$ be the Dihedral group, and $\kappa $ an $n$-dimensional unitary representation of $D_{2n}$ acting in $\mathbb{C}^n$ as follows. $(\kappa (r)v)(j)=v((j-1)\mod n)$ and $(\kappa(s)v)(j)=v((n-j)\mod n)$ for $v\in\mathbb{C}^n.$ For any representation which is unitarily equivalent to $\kappa,$ we prove that when $n$ is prime there exists a Zariski open subset $E$ of $\mathbb{C}^{n}$ such that for any vector $v\in E,$ any subset of cardinality $n$ of the orbit of $v$ under the action of this representation is a basis for $\mathbb{C}^{n}.$ However, when $n$ is even there is no vector in $\mathbb{C}^{n}$ which satisfies this property. As a result, we derive that if $n$ is prime, for almost every (with respect to Lebesgue measure) vector $v$ in $\mathbb{C}^{n}$ the $\Gamma $-orbit of $v$ is a frame which is maximally robust to erasures. We also consider the case where $\tau$ is equivalent to an irreducible unitary representation of the Dihedral group acting in a vector space $\mathbf{H}_{\tau}\in\left\{\mathbb{C},\mathbb{C}^2\right\}$ and we provide conditions under which it is possible to find a vector $v\in\mathbf{H}_{\tau}$ such that $\tau\left( \Gamma\right) v$ has the Haar property.