# Subfactors and Hadamard Matrices

## TL;DR

This paper explores the connection between complex Hadamard matrices and subfactors, analyzing their symmetries and invariants, and examining specific cases like Dita type matrices to reveal additional structural features.

## Contribution

It introduces a method to associate subfactors to Hadamard matrices and investigates their invariants, especially for small dimensions and Dita type matrices, uncovering new structural insights.

## Key findings

- Computed initial relative commutants for small-dimensional Hadamard matrices.
- Identified intermediate subfactors in Dita type matrices.
- Revealed additional structure in standard invariants beyond Jones projections.

## Abstract

To any complex Hadamard matrix H one associates a spin model commuting square, and therefore a hyperfinite subfactor. The standard invariant of this subfactor captures certain "group-like" symmetries of H. To gain some insight, we compute the first few relative commutants of such subfactors for Hadamard matrices of small dimensions. Also, we show that subfactors arising from Dita type matrices have intermediate subfactors, and thus their standard invariants have some extra structure besides the Jones projections.