Permutation equivalence classes of kronecker products of unitary Fourier matrices

Kronecker products of unitary Fourier matrices play an 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 the 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.