The Projective Symmetry Group of a Finite Frame

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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 \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.

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