The Fourier transform of a projective group frame (Q2175014)

A projective group frame is the orbit \(\rho(G)v\) of a nonzero vector \(v\) under a projective representation \(\rho\) of a finite group \(G\). These are stated to be of interest in many areas -- e.g. mutually unbiased bases, spherical \(t\)-designs etc. The present paper studies them via the Gramian matrices of the representations. A complete description of the projective group frames in terms of the irreducible projective representations of \(G\) is known. The problem considered here is that of starting with 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 \(v\). Given a Gramian \(P\), the author gives a concrete construction of a projective group frame whose associated Gramian matrix is \(P\). In the author's words: ``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.'' The Fourier transform for projective representations is studied for this purpose. For any cocycle \(\alpha\) on \(G\), the author defines a \((G,\alpha)\)-matrix and it is shown that the set of such matrices form a \(C^*\)-algebra \(M_{(G,\alpha)}\). A unique Fourier decomposition of group matrices is given. It is proved that the Gramian determines a projective group frame up to unitary equivalence and shows that tight projective group frames correspond to projections. The Klein four-group and the dihedral groups are discussed as examples.