Title

Dihedral Group Frames which are Maximally Robust to Erasures

Publication Date

2014

Document Type

Article

Abstract

Let n be a natural number larger than two. Let D2n = ⟨r, s: rn = s2 = e, srs = rn1 ⟩ be the Dihedral group, and κ an n-dimensional unitary representation of D2n acting in ℂn as follows. (κ(r)v)(j)=v((j − 1) mod n) and (κ(s)v)(j) = v((nj) mod n) for v ∈ ℂn. For any representation which is unitarily equivalent to κ, we prove that when n is prime there exists a Zariski open subset E of ℂn such that for any vector vE, any subset of cardinality n of the orbit of v under the action of this representation is a basis for ℂn. However, when n is even there is no vector in ℂn 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 ℂn 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, 2 } and we provide conditions under which it is possible to find a vector v ∈ Hτ such that τ(Γ)vhas the Haar property.

Comments

This is the pre-print version harvested from ArXiv. (http://arxiv.org/abs/1408.2022).

Published version available:

Oussa, V. (2015). Dihedral Group Frames which are Maximally Robust to Erasures. Linear and Multilinear Algebra, 63(12), 2494-2510. doi: 10.1080/03081087.2015.1020746

Original Citation

Oussa, V. (2014). Dihedral Group Frames which are Maximally Robust to Erasures. arXiv:1408.2022 [math.RT].