Group frames

Shayne Waldron


The prototypical example of a tight frame: the Mercedes--Benz frame can be obtained as the orbit of a single vector under the action of the group generated by rotation by 2*pi/3, or the dihedral group of symmetries of the triangle. Many frames used in applications are constructed in this way, often as the orbit of a single vector (akin to a mother wavelet). Most notable are the harmonic frames (finite abelian groups) used in signal analysis, and the equiangular Heisenberg frames, or SIC-POVMs, (discrete Heisenberg group) used in quantum information theory. Other examples include tight frames of multivariate orthogonal polynomials sharing symmetries of the weight function, and the highly symmetric tight frames which can be viewed as the vertices of highly regular polytopes. We will describe the basic theory of such group frames, and some of the constructions that have been found so far.

Keywords: Group frame, G-frame, harmonic frames, SIC-POVM, Heisenberg frame, highly symmetric tight frame, symmetry group of a frame, Heisenberg frame, group matrix, unitary representation, equiangular frames, Zauner's conjecture

Math Review Classification: Primary 42C15, 51F15, 52B11; Secondary 20F55, 51M30, 52B15

Length: 20 pages

Last Updated: 4 Jan 2012