Dihedral Group Frames Which Are Maximally Robust to Erasures

Let n be a natural number larger than two. Let D2n = ⟨r, s ∶ r = s = e, srs = r⟩ be the Dihedral group, and κ an n-dimensional unitary representation of D2n acting in C as follows. (κ(r)v)(j) = v((j−1) mod n) and (κ(s)v)(j) = v((n − j) mod n) for v = (v0,⋯, vn−1) ∈ C. For any representation which is unitarily equivalent to κ, we prove that when n is prime there exists a Zariski open subset E of C such that for any vector v ∈ E, any subset of cardinality n of the orbit of v under the action of this representation is a basis for C. However, when n is even there is no vector in C 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 C the Γ-orbit of v is a frame which is maximally robust to erasures. We also consider the case where τ is equivalent to an irreducible unitary representation of the Dihedral group acting in a vector space Hτ ∈ {C,C} and we provide conditions under which it is possible to find a vector v ∈Hτ such that τ (Γ) v has the Haar property.