Homomorphisms of matrix algebras and constructions of Butson–Hadamard​ matrices

Abstract

An n×n matrix H is Butson–Hadamard if its entries are kth roots of unity and it satisfies HH∗=nIn. Write BH(n,k) for the set of such matrices.Suppose that k=pαqβ where p and q are primes and α≥1. A recent result of Östergård and Paavola uses a matrix H∈BH(n,pk) to construct H′∈BH(pn,k). We simplify the proof of this result and remove the restriction on the number of prime divisors of k. More precisely, we prove that if k=mt, and each prime divisor of k divides t, then we can construct a matrix H′∈BH(mn,t) from any H∈BH(n,k).