A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{ \ast} = nI_{ n}$. We write $\mathrm{ BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\mathrm{ BH}(n, k) \rightarrow \mathrm{ BH}(m, \ell)$. In this paper, we develop a technique for controlling the spectra of certain Hadamard matrices. For each integer $t$, we construct a real Hadamard matrix $H_{ t}$ of order $n_{ t} = 2^{ 2^{ t-1}-1}$ such that the minimal polynomial of $\frac{ 1}{ \sqrt{ n_{ t}}}H_{ t}$ is the cyclotomic polynomial $\Phi_{ 2^{ t+1}}(x)$. Such matrices yield new examples of complete morphisms \[ \mathrm{ BH}(n, 2^{ t}) \rightarrow \mathrm{ BH}(2^{ 2^{ t-1}-1}n, 2)\,, \] for each $t \geq 2$, generalising a well-known result of Turyn.
Categories: math.CO