The defect of generalized Fourier matrices

The N × N complex Hadamard matrices form a real algebraic manifold CN. We have , and following Tadej and Życzkowski we investigate here the computation of the enveloping tangent space , and notably of its dimension , called undephased defect of H. Our main result is an explicit formula for the defect of the Fourier matrix FG associated to an arbitrary finite abelian group G=ZN1×⋯×ZNr. We also comment on the general question “does the associated quantum permutation group see the defect”, with a probabilistic speculation involving Diaconis–Shahshahani type variables.