# On the construction of highly symmetric tight frames and complex polytopes

## Helen Broome and Shayne Waldron

## Abstract:

Many ``highly symmetric'' configurations of vectors in C^d,
such as the vertices
of the platonic solids and the regular complex polytopes, are
*equal-norm tight frames* by virtue of being the
orbit of the irreducible unitary action of their symmetry group.
For nonabelian groups there are *uncountably* many
such tight frames up to unitary equivalence.
The aim of this paper is to single out those orbits
which are particularly nice,
such as those which are the vertices of a complex polytope.
This is done by defining a *finite*
class of tight frames of n vectors for C^d (n and d fixed)
which we call the *highly symmetric tight frames*.
We outline how these frames can be calculated from the representations of abstract groups
using a computer algebra package.
We give numerous examples, with a special emphasis on
those obtained from the (Shephard-Todd) finite reflection groups.
The interrelationships between these frames with complex polytopes,
harmonic frames, equiangular tight frames,
and Heisenberg frames (maximal sets of equiangular lines) are explored in detail.

**Keywords:**
(unitary) reflection,
pseudoreflection,
(Shephard-Todd) finite reflection group,
regular complex polytopes,
line systems,
symmetry groups,
harmonic frames,
equal-norm tight frames,
equiangular tight frames,
Monster group

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

**Length:** xx pages

**Last Updated:** 10 May 2010

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