A finiteness result for complex Hadamard matrices in maximal Abelian self-adjoint subalgebras of $\rm M_n (\mathbb{C})$

Abstract

Let $\mathcal A=U \rm D_n(\mathbb C)$$U^*$ be a Maximal Abelian Self-Adjoint subalgebra (MASA) of $\rm M_n(\mathbb C)$, where $\rm D_n(\mathbb C)$ denotes the diagonal matrices and $U\in \rm M_n(\mathbb C)$ is a unitary matrix. Assume that $U$ is full superregular, i.e., all the minors of $U$ are nonzero. We show that $\mathcal A$ contains at most finitely many complex Hadamard matrices, up to equivalence given by multiplication by complex units. In particular, since almost every unitary is full superregular (with respect to the Haar distribution), it follows that almost every MASA of $\rm M_n(\mathbb C)$ contains only finitely many complex nonequivalent Hadamard matrices.