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.

通过其Gramian矩阵可构造许多关注的紧框架（该框架确定直到等价的框架）。给定这样的格拉姆准则，可以确定紧框架是否是射影群框架，即，是否是某个群G的射影轨道（可能不是唯一的）。另一方面，根据G的不可约投影表示，对投影群框架有完整的描述。在这里，我们考虑一个反问题，即对一个群G取一个射影群框架的Gramian ，并识别余弦并明确地将该框架构造为向量v的射影群轨道（根据不可约数分解）。关键思想是要认识到Gramian是由向量给定的群矩阵$F\in {\mathbb{C}}^G$，然后对f进行傅立叶变换，得到v的分量作为正交投影。

这需要发展群矩阵理论和用于投影表示的傅立叶变换。特别令人感兴趣的是，我们给出了（射影）群矩阵的块对角化。这导致组矩阵的唯一傅里叶分解，并进一步细分为低秩组矩阵。