# The symmetry group of a finite frame

## Richard Vale and Shayne Waldron

## Abstract:

We define the *symmetry group* of a finite frame
as a group of permutations on its index set.
This group is closely related to the symmetry group of [VW05]
for *tight* frames:
they are isomorphic when the frame is tight and has distinct vectors.
The symmetry group is the same for all similar frames,
in particular for a frame, its dual and canonical tight frames.
It can easily be calculated from the Gramian matrix of the
canonical tight frame.
Further, a frame and its complementary frame have the
same symmetry group. We exploit this last property to
construct and and classify some classes of highly symmetric
tight frames.

**Keywords:**
finite frame,
geometrically uniform frame,
Gramian matrix,
harmonic frame,
maximally symmetric frames
partition frames,
symmetry group,
tight frame,

**Math Review Classification:**
Primary 42C15, 58D19;
Secondary 42C40, 52B15

**Length:** 17 pages

**Last Updated:** 17 September 2009

## Availability: