Morphisms of Butson classes

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 $\ell < k$.