Let q be a power of 2. We show by representation theory that there exists a q
x q unitary matrix of multiplicative order q+1 whose powers generate q+1
pairwise mutually unbiased base in C^q. When q is a power of an odd prime,
there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2
powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the
existence of these matrices implies the existence of a special type of
orthogonal decomposition with respect to the Killing form of the special linear
and symplectic Lie algebras.