A noncommutative construction of families of biunitary matrices and application to subfactors

TL;DR

This paper presents a novel noncommutative method to construct families of biunitary matrices from complex Hadamard matrices, leading to new subfactors with rich automorphism structures and explicit classifications.

Contribution

It introduces a noncommutative construction of biunitary matrices from pairs of Hadamard matrices, producing new subfactors and detailed automorphism analysis.

Findings

Generated infinite families of biunitary matrices of varying orders.

Constructed nested sequences of vertex model subfactors not arising from basic towers.

Provided explicit classifications and automorphism structures for subfactors related to Fourier matrices.

Abstract

We introduce a construction that, given a pair (u,v) of complex Hadamard matrices of the same order, generates infinitely many biunitary matrices of varying (and distinct) orders. As a key application, this framework yields nested sequences of vertex model subfactors that are not a tower of downward basic construction. Notably, the construction is noncommutative: interchanging the matrices (i.e., considering (v,u) instead of (u,v)) can lead to non-isomorphic subfactors. Focusing on the Hadamard equivalence class of the Fourier matrix, we provide a full characterization of the resulting vertex model subfactors, along with explicit computations of their relative commutants. Along the way, we conduct a detailed study of certain naturally arising inner and outer automorphisms that play a key role in the structure of these subfactors.