Complete Set of Unitary Irreps of Discrete Heisenberg Group \( HW_{2^{s}}\)

Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group \(HW_{2^{s}}\) over the discrete phase space lattice \(\mathbb {Z}_{2^{s}}\otimes \) \(\mathbb {Z}_{2^{s}}\) is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.