The defect of generalized Fourier matrices

TL;DR

This paper investigates the geometric structure of complex Hadamard matrices, providing an explicit formula for the defect of Fourier matrices linked to finite abelian groups, and explores implications for quantum permutation groups.

Contribution

It derives an explicit formula for the defect of Fourier matrices associated with finite abelian groups, advancing understanding of their geometric and algebraic properties.

Findings

Explicit formula for the defect of Fourier matrices

Analysis of the tangent space dimension of complex Hadamard matrices

Discussion on the quantum permutation group and defect relationship

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.