The Fourier transform of a projective group frame

TL;DR

This paper develops a Fourier transform approach for analyzing and explicitly constructing projective group frames from their Gramian matrices, enabling decomposition into irreducible components and low-rank structures.

Contribution

It introduces a Fourier transform framework for projective group frames, allowing explicit construction and decomposition from Gramian matrices using group matrix theory.

Findings

Provides a block diagonalisation of projective group matrices

Establishes a unique Fourier decomposition for these matrices

Enables decomposition into low-rank components

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 $f\in\mathbb{C}^G$, 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.