# Tight frames generated by finite nonabelian groups

## Richard Vale and Shayne Waldron

## Abstract:

Let $\cH$ be a Hilbert space of finite dimension $d$,
such as the finite signals $\Cd$
or a space of multivariate orthogonal polynomials, and
$n\ge d$. There is a finite number of tight frames of $n$ vectors
for $\cH$ which can be obtained as the orbit of a single vector
under the unitary action of an abelian group $G$ (of symmetries of the frame).
Each of these so called {\it harmonic frames} or {\it geometrically
uniform frames} can be obtained from the
character table of $G$ in a simple way.
These frames are used in signal processing and information
theory.
For a nonabelian group $G$ there are in general uncountably
many inequivalent tight frames of $n$ vectors for $\cH$ which
can be obtained as such a $G$--orbit.
However,
by adding an additional natural symmetry condition
(which automatically holds if $G$ is abelian), we obtain a finite
class of such frames which can be constructed from the character table
of $G$ in a similar fashion to the harmonic frames.
This is done by identifying each $G$--orbit with an element of
the group algebra $\CC G$ (via its Gramian), imposing the condition
in the group algebra,
and then describing the corresponding class of tight frames.

**Keywords:**
signal processing,
information theory,
finite nonabelian groups,
representation theory,
group matrices,
tight frames,
harmonic frames,
geometrically uniform frames,
Gramian matrix,
central tight $G$--frames

**Math Review Classification:**
Primary 20C15, 42C15, 41A63, 41A65;
Secondary 65T60, 94A11, 94A12, 94A15

**Length:** 15 pages

**Last Updated:** 18 July 2007

## Availability: