Homomorphisms of matrix algebras and constructions of Butson-Hadamard matrices

An n x n matrix H is Butson-Hadamard if its entries are kth roots of unity and it satisfies HH* = nI(n). Write BH(n, k) for the set of such matrices. Suppose that k = p(alpha)q(beta )where p and q are primes and alpha >= 1. A recent result of Ostergard and Paavola uses a matrix H is an element of BH(n, pk) to construct H' is an element of 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' is an element of BH(mn, t) from any H is an element of BH(n, k). (C) 2019 Elsevier B.V. All rights reserved.