First order deformations of the Fourier matrix

TL;DR

This paper investigates the structure of deformations of the Fourier matrix within the space of complex Hadamard matrices, providing a clear description of the first order deformation cone and confirming a rationality conjecture at this point.

Contribution

It offers a simple description of the first order deformation cone of the Fourier matrix and verifies the rationality conjecture for this specific case.

Findings

Explicit description of the first order deformation cone for the Fourier matrix

Validation of the rationality conjecture at the Fourier matrix point

Enhanced understanding of the local structure of complex Hadamard matrices

Abstract

The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. The singularity at a point $H\in C_N$ is described by a filtration of cones $T^\times_HC_N\subset T^\circ_HC_N\subset T_HC_N\subset\widetilde{T}_HC_N$, coming from the trivial, affine, smooth and first order deformations. We study here these cones in the case where $H=F_N$ is the Fourier matrix, $(w^{ij})$ with $w=e^{2\pi i/N}$, our main result being a simple description of $\widetilde{T}_HC_N$. As a consequence, the rationality conjecture $dim_\mathbb R(\widetilde{T}_HC_N)=dim_\mathbb Q(\widetilde{T}_HC_N\cap M_N(\mathbb Q))$ holds at $H=F_N$.