# The Fourier transform of a projective group frame

## Abstract:

Many tight frames of interest are constructed via their Gramian matrix (which determines the frame up to unitary equivalence). Given such a Gramian, it can be determined whether or not the tight frame is projective group frame, i.e., is the projective orbit of some group $G$ (which may not be unique). On the other hand, there is complete description of the projective group frames in terms of the irreducible projective representations of $G$. Here we consider the inverse problem of taking the Gramian of a projective group frame for a group $G$, and identifying the cocycle and constructing the frame explicitly as the projective group orbit of a vector $v$ (decomposed in terms of the irreducibles). The key idea is to recognise that the Gramian is a group matrix given by a vector $f\in\CC^G$, and to take the Fourier transform of $f$ to obtain the components of $v$ as orthogonal projections. This requires the development of a theory of group matrices and the Fourier transform for projective representations. Of particular interest, we give a block diagonalisation of (projective) group matrices. This leads to a unique Fourier decomposition of the group matrices, and a further fine-scale decomposition into low rank group matrices.

Keywords: $(G,\alpha)$-frame, Group matrix, Gramian matrix, projective group frame, twisted group frame, central group frame, Fourier transform, character theory, Schur multiplier, projective representation.

Math Review Classification: Primary 20C15, 20C25, 42C15, 43A32, 65T50; Secondary 05B30, 42C40, 43A30, 94A12.

Length: 26 pages

Last Updated: 14 June 2018