Morphisms of Butson classes

TL;DR

This paper introduces morphisms between Butson Hadamard matrices over different roots of unity, generalizing previous constructions and classifying specific morphisms, thus advancing the understanding of their algebraic structure.

Contribution

It defines and constructs morphisms between Butson matrices over different roots of unity, unifying and extending prior results in the field.

Findings

Constructed tensor-product-like morphisms reducing root order

Unified Turyn's and Compton-Craigen-de Launey's constructions

Classified morphisms from $n imes n$ to $2n imes 2n$ matrices

Abstract

We introduce the concept of a morphism from the set of Butson Hadamard matrices over kth roots of unity to the set of Butson matrices over $\ell$th roots of unity. As concrete examples of such morphisms, we describe tensor-product-like maps which reduce the order of the roots of unity appearing in a Butson matrix at the cost of increasing the dimension. Such maps can be constructed from Butson matrices with eigenvalues satisfying certain natural conditions. Our work unifies and generalises Turyn's construction of real Hadamard matrices from Butson matrices over the 4th roots and the work of Compton, Craigen and de Launey on `unreal' Butson matrices over the 6th roots. As a case study, we classify all morphisms from the set of $n \times n$ Butson matrices over kth roots of unity to the set of $2n\times 2n$ Butson matrices over $\ell$th roots of unity where $\ell < k$.