# The projective symmetry group of a finite frame

## Tuan-Yow Chien and Shayne Waldron

## Abstract:

We define the projective symmetry group of a finite sequence of vectors
(a frame) in a natural way as a group of permutations on the vectors
(or their indices). This definition ensures that the projective symmetry
group is the same for a frame and its complement. We give a parallelisable
algorithm for computing the projective symmetry group from a small set of
projective invariants when the underlying field is a subfield of $\C$
which is closed under conjugation. This algorithm is applied in a number
of examples including equiangular lines (in particular SICs), MUBs
and harmonic frames.

**Keywords:**
Projective unitary equivalence, Gramian, Gram matrix, harmonic frame,
equiangular tight frame, SIC-POVM (symmetric informationally complex positive operator
valued measure), MUB (mutually orthogonal bases), triple products, Bargmann invariants,
projective symmetry group

**Math Review Classification:**
Primary 20C25, 42C15, 81P15, 94A15;
Secondary 11L03, 14N20, 20C15, 52B11

**Length:** 30 pages

**Last Updated:** 16 August 2018

## Availability: