The projective symmetry group of a finite frame

Tuan-Yow Chien and Shayne Waldron


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