  We show that naturally associated to a SIC (symmetric informationally
complete positive operator valued measure or SIC-POVM) in dimension d there are
a number of higher dimensional structures: specifically a projector and a
complex Hadamard matrix in dimension d squared and a pair of ETFs (equiangular
tight frames) in dimensions d(d-1)/2, d(d+1)/2. We also show that a WH (Weyl
Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair
of symmetric tight fusion frames in dimension d. We deduce two relaxations of
the WH SIC existence problem. We also find a reformulation of the problem in
which the number of equations is fewer than the number of variables. Finally,
we show that in at least four cases the structures associated to a SIC lie on
continuous manifolds of such structures. In two of these cases the manifolds
are non-linear. Restricted defect calculations are consistent with this being
true for the structures associated to every known SIC with d between 3 and 16,
suggesting it may be true for all d greater than 2.
