On complete morphisms of Butson Hadamard matrices
A complete morphism of Butson classes replaces each entry of a matrix in $\mathrm{BH}(n,k)$ by an $m \times m$ block, uniformly, to produce a matrix in $\mathrm{BH}(mn,t)$. We show that every complete morphism is described by a pair $(U,Z)$, where $Z \in \mathrm{BH}(m,t)$ and $U$ is a unitary matrix annihilated by the $k^{\textrm{th}}$ cyclotomic polynomial. We deduce from a theorem of Lam and Leung that complete morphisms onto real Hadamard matrices exist precisely when $k$ is a power of $2$. The exponent-reducing morphisms of Östergård--Paavola and Ó Catháin--Swartz achieve the smallest block size the degree bound allows, and we show that there are no other complete morphisms for which $U$ is monomial. We construct new complete morphisms where $U$ is not monomial: a complete morphism $\mathrm{BH}(n,7) \rightarrow \mathrm{BH}(6n,6)$ is given, the first to eliminate a prime from the alphabet.