On Complete Morphisms of Butson Hadamard Matrices

ABSTRACT A complete morphism of Butson classes replaces each entry of a matrix in by an block, uniformly, to produce a matrix in . We show that every complete morphism is described by a pair , where and is a unitary matrix annihilated by the cyclotomic polynomial. We deduce from a theorem of Lam and Leung that complete morphisms onto real Hadamard matrices exist precisely when 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 is monomial. We construct new complete morphisms where is not monomial: a complete morphism is given, the first to eliminate a prime from the alphabet.