#### 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 *D*_{2}_{n} = ⟨*r*, *s*: *r*^{n} = s^{2} = *e*, *srs* = *r*^{n}^{−}* ^{1}* ⟩ be the Dihedral group, and κ an

*n*-dimensional unitary representation of

*D*

_{2}

_{n}acting in ℂ

^{n}as follows. (κ(

*r*)

*v*)(

*j*)=

*v*((

*j*− 1) mod

*n*) and (κ(

*s*)

*v*)(

*j*) =

*v*((

*n*−

*j*) mod

*n*) for

*v*∈ ℂ

^{n}. 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

^{n}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

^{n}. However, when

*n*is even there is no vector in 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 ℂ

^{n}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 H

_{τ}∈{ ℂ

^{, }ℂ

^{2}} and we provide conditions under which it is possible to find a vector

*v*∈ H

_{τ}such that τ(Γ)

*v*has the Haar property.

#### Original Citation

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

#### Virtual Commons Citation

Oussa, Vignon (2014). Dihedral Group Frames which are Maximally Robust to Erasures. In *Mathematics Faculty Publications.* Paper 43.

Available at: http://vc.bridgew.edu/math_fac/43

## 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