The symmetry group of a finite frame

Richard Vale and Shayne Waldron


We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of [VW05] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and and classify some classes of highly symmetric tight frames.

Keywords: finite frame, geometrically uniform frame, Gramian matrix, harmonic frame, maximally symmetric frames partition frames, symmetry group, tight frame,

Math Review Classification: Primary 42C15, 58D19; Secondary 42C40, 52B15

Length: 17 pages

Last Updated: 17 September 2009