Homomorphisms of matrix algebras and constructions of Butson-Hadamard matrices

Abstract

An $n\times n$ matrix $H$ is Butson-Hadamard if its entries are $k^{\text{th}}$ roots of unity and it satisfies $H{H}^{\ast }=n{I}_n$. Write $BH(n,k)$ for the set of such matrices. Suppose that $k=p^{\alpha }{q}^{\beta }$ where $p$ and $q$ are primes and $\alpha \ge 1$. A recent result of {Ö}stergård and Paavola uses a matrix $H\in BH(n,pk)$ to construct $H^{\prime }\in BH(pn,k)$. 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 $H^{\prime }\in BH(mn,t)$ from any $H\in BH(n,k)$.

5 pages