Kronecker products of unitary Fourier matrices play important role in solving
multilevel circulant systems by a multidimensional Fast Fourier Transform. They
are also special cases of complex Hadamard (Zeilinger) matrices arising in many
problems of mathematics and theoretical physics. The main result of the paper
is splitting the set of all kronecker products of unitary Fourier matrices into
permutation equivalence classes. The choice of permutation equivalence to
relate the products is motivated by the quantum information theory problem of
constructing maximally entangled bases of finite dimensional quantum systems.
Permutation inequivalent products can be used to construct inequivalent, in a
certain sense, maximally entangled bases.Comment: 26 page