Intersection of conjugate Hadamard subfactors arising from Fourier matrices
Keshab Chandra Bakshi, Satyajit Guin, Guruprasad
TL;DR
The paper resolves how two conjugate Hadamard subfactors from the same Fourier-tensor class can intersect inside the hyperfinite II$_1$ factor. By combining vertex-model and Hadamard-subfactor machinery with crossed-product realizations, it derives an explicit finite Jones index for the intersection, $[R:R_U\cap R_V]=N^2/|H|$, where $|H|=\dim(\mathrm{Ad}_U(\Delta_N)\cap \mathrm{Ad}_V(\Delta_N))$, and identifies the relative commutant as $(\mathrm{Ad}_U(\Delta_N)\cap \mathrm{Ad}_V(\Delta_N))'\cap \mathbb{C}^N$. A sharp criterion determines when the intersection is a vertex model subfactor, and the paper shows that every admissible index of the form $N^2/|H|$ occurs. It also provides an exact Connes-Størmer entropy formula $h(R_U|R_V)=\frac{1}{N}\sum_{i,j}|(U^*V)_{ij}|^2\eta$, linking entropic behavior to the overlap of diagonal algebras via $U^*V$. Together, these results reveal how the internal algebraic structure of complex Hadamard matrices governs both the positional geometry and entropy of Hadamard intersections.
Abstract
Given two distinct complex Hadamard matrices belonging to the same equivalence class generated by the tensor products of Fourier matrices, we show that if the corresponding Hadamard subfactors are conjugate, then their intersection is a factor with finite Jones index. We compute the index of the intersection explicitly and determine its relative commutant. Furthermore, we precisely characterize when these intersections give rise to vertex model subfactors, thereby extending our earlier results in low dimensions. As an application, we derive an explicit formula for the Connes-Størmer relative entropy associated with these intersections. These results reveal how the internal algebraic structure of complex Hadamard matrices governs the relative position and entropic behaviour of the subfactors.