The defect of generalized Fourier matrices

The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. We have $C_N=M_N(\mathbb{T})\cap \sqrt{N}{U}_N$, and following Tadej and .Zyczkowski we investigate here the computation of the enveloping tangent space ${\tilde{T}}_H{C}_N=T_H{M}_N(\mathbb{T})\cap T_H\sqrt{N}{U}_N$, and notably of its dimension $d(H)=\dim ({\tilde{T}}_H{C}_N)$, called undephased defect of $H$. Our main result is an explicit formula for the defect of the Fourier matrix $F_G$ associated to an arbitrary finite abelian group $G=\mathbb Z_{N_1}\times…\times\mathbb Z_{N_r}$. 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.