Generation of Mutually Unbiased Bases as Powers of a Unitary Matrix in 2-power Dimensions

Let q be a power of 2. We show by representation theory that there exists a q × q unitary matrix of multiplicative order q + 1 whose powers generate q + 1 complex pairwise mutually unbiased bases in C. When q is a power of an odd prime, there is a q × q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 complex pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of special orthogonal decompositions of certain simple Lie algebras.