Journal of Operator Theory

Volume 95, Issue 2, Spring 2026 pp. 537-566.

On the structure of the moduli space of commuting squares around the Fourier spin model

Authors: Shuler Hopkins (1), Remus Nicoară (2)
Author institution: (1) Department of Mathematics, Sewanee: The University of the South, Sewanee, TN, 37383, U.S.A.
(2) Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, U.S.A.

Summary: We investigate the structure of the space of commuting squares around the Fourier spin model commuting square, or equivalently the structure of the space of complex $n\times n$ Hadamard matrices around the Fourier matrix $F_n$, by finding restrictions for the possible directions of tangency at $F_n$. As an application, we show that for $n=30$ the dimension of any differentiable family of complex Hadamard matrices containing $F_{n}$ is strictly less than the dimension of the enveloping tangent space at $F_{n}$ (called the defect of $F_{n}$).

DOI: http://dx.doi.org/10.7900/jot.2024sep20.2508
Keywords: commuting squares, complex Hadamard matrices, Fourier matrix

Contents Full-Text PDF