Hermitian unitary matrices with modular permutation symmetry

Abstract

We study Hermitian unitary matrices $\mathcal{S}\in {\mathbb{C}}^{n,n}$ with the following property: There exist $r\ge 0$ and $t>0$ such that the entries of $\mathcal{S}$ satisfy $|{\mathcal{S}}_{jj}|=r$ and $|{\mathcal{S}}_{jk}|=t$ for all $j,k=1,\dots ,n$, $j\ne k$. We derive necessary conditions on the ratio $d:=r/t$ and show that these conditions are very restrictive except for the case when $n$ is even and the sum of the diagonal elements of $\S$ is zero. Examples of families of matrices $\mathcal{S}$ are constructed for $d$ belonging to certain intervals. The case of real matrices $\mathcal{S}$ is examined in more detail. It is demonstrated that a real $\mathcal{S}$ can exist only for $d=\frac{n}{2}-1$, or for $n$ even and $\frac{n}{2}+d\equiv 1(\mathrm{mod}2)$. We provide a detailed description of the structure of real $\mathcal{S}$ with $d\ge \frac{n}{4}-\frac{3}{2}$, and derive a sufficient and necessary condition of their existence in terms of the existence of certain symmetric $(v,k,\lambda )$-designs. We prove that there exist no real $\mathcal{S}$ with $d\in \left(\frac{n}{6}-1,\frac{n}{4}-\frac{3}{2}\right)$. A parametrization of Hermitian unitary matrices is also proposed, and its generalization to general unitary matrices is given. At the end of the paper, the role of the studied matrices in quantum mechanics on graphs is briefly explained.

revised version, 21 pages

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Cited by in corpus (5)

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• A generalization of circulant Hadamard and conference matrices
• Manakov system on metric graphs: Modeling the reflectionless propagation of vector solitons in networks
• Quantum graph vertices with permutation-symmetric scattering probabilities
• Inverse scattering problem for quantum graph vertices