# The Projective Symmetry Group of a Finite Frame

### 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

Shayne Waldron

Department of Mathematics

University of Auckland

Private Bag 92019, Auckland, New Zealand.

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 $\mathbb{C}$ 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.

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