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
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.