The defect of generalized Fourier matrices

Abstract

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 $\widetilde{T}_HC_N=T_HM_N(\mathbb T)\cap T_H\sqrt{N}U_N$, and notably of its dimension $d(H)=\dim(\widetilde{T}_HC_N)$, 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.