# Group frames

## Shayne Waldron

## Abstract:

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

## Availability: