# The Projective Symmetry Group of a Finite Frame

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 48, (2018), **Pages** 55-81

**Tuan-Yow Chien**

Department of Mathematics

University of Auckland

Private Bag 92019, Auckland, New Zealand

mailto:tchi043@ec.auckland.ac.nz

**Shayne Waldron**

Department of Mathematics

University of Auckland

Private Bag 92019, Auckland, New Zealand.

mailto:waldron@math.auckland.ac.nz

**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 an
algorithm for computing the projective symmetry group
from a small set of projective invariants when the
underlying field is a subfield of 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.

**Classification** (MSC2000) Primary: 20C25, 42C15, 81P15, 94A15. Secondary: 11L03, 14N20, 20C15. 52B11.