First order deformations of the Fourier matrix

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 $T_H^{\times }{C}_N\subset T_H^{\circ }{C}_N\subset T_H{C}_N\subset {\tilde{T}}_H{C}_N$, 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 ${\tilde{T}}_H{C}_N$. As a consequence, the rationality conjecture $di{m}_{\mathbb{R}}({\tilde{T}}_H{C}_N)=di{m}_{\mathbb{Q}}({\tilde{T}}_H{C}_N\cap M_N(\mathbb{Q}))$ holds at $H=F_N$.

26 pages

References in corpus (2)

• A finiteness result for commuting squares of matrix algebras
• Defect and equivalence of unitary matrices. The Fourier case

Cited by in corpus (3)

• Defect and equivalence of unitary matrices. The Fourier case
• Regularity questions for complex Hadamard matrices
• The Gale-Berlekamp game for Hadamard matrices