# The Projective Symmetry Group of a Finite Frame

## 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## 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.

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## Published

2018-12-31

## How to Cite

*New Zealand Journal of Mathematics*,

*48*, 55–81. Retrieved from https://nzjmath.org/index.php/NZJMATH/article/view/35

## Issue

## Section

Articles