Fusion algebras for imprimitive complex reflection groups

We prove that the Fourier matrices for the imprimitive complex reflection groups introduced by Malle in [Gunter Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 (3) (1995) 768–826] define fusion algebras with not necessarily positive but integer structure constants. Hence they define Z-algebras. As a result, we obtain that all known Fourier matrices belonging to spetses define algebras with integer structure constants.