Dihedral group frames which are maximally robust to erasures
Let be a natural number larger than two. Let be the Dihedral group, and an -dimensional unitary representation of acting in as follows. and for . For any representation which is unitarily equivalent to we prove that when is prime, there exists a Zariski open subset of such that for any vector any subset of cardinality of the orbit of under the action of this representation is a basis for . However, when is even, there is no vector in which satisfies this property. As a result, we derive that if is prime, for almost every (with respect to Lebesgue measure) vector in , the -orbit of is a frame which is maximally robust to erasures. We also consider the case where is equivalent to an irreducible unitary representation of the Dihedral group acting in a vector space and we provide conditions under which it is possible to find a vector such that has the Haar property.