# The Fourier transform of a projective group frame

## Shayne Waldron

## Abstract:

Many tight frames of interest are constructed via their Gramian matrix
(which determines the frame up to unitary equivalence).
Given such a Gramian, it can be determined whether or not the tight
frame is projective group frame, i.e., is the projective orbit of some
group $G$ (which may not be unique).
On the other hand, there is complete description of the projective group
frames in terms of the irreducible projective representations of $G$.
Here we consider the inverse problem of taking the Gramian of a projective
group frame for a group $G$, and identifying the cocycle
and constructing the frame explicitly as
the projective group orbit of a vector $v$ (decomposed in terms of
the irreducibles). The key idea is to recognise that the Gramian
is a group matrix given by a vector $f\in\CC^G$, and
to take the Fourier transform of $f$ to obtain
the components of $v$ as orthogonal projections.
This requires the development of a theory of
group matrices and the Fourier transform for projective representations.
Of particular interest, we give a block diagonalisation of (projective) group matrices.
This leads to a unique Fourier decomposition of the group matrices,
and a further fine-scale decomposition into low rank group matrices.

**Keywords:**
$(G,\alpha)$-frame,
Group matrix,
Gramian matrix,
projective group frame,
twisted group frame,
central group frame,
Fourier transform,
character theory,
Schur multiplier,
projective representation.

**Math Review Classification:**
Primary 20C15, 20C25, 42C15, 43A32, 65T50;
Secondary 05B30, 42C40, 43A30, 94A12.

**Length:** 26 pages

**Last Updated:** 14 June 2018

## Availability: