  We generalize previous results on N=1, (3+1)-dimensional superconformal block
quiver gauge theories. It is known that the necessary conditions for a theory
to be superconformal, i.e. that the beta and gamma functions vanish in addition
to anomaly cancellation, translate to a Diophantine equation in terms of the
quiver data. We re-derive results for low block numbers revealing an new
intriguing algebraic structure underlying a class of possible superconformal
fixed points of such theories. After explicitly computing the five block case
Diophantine equation, we use this structure to reorganize the result in a form
that can be applied to arbitrary block numbers. We argue that these theories
can be thought of as vectors in the root system of the corresponding quiver and
superconformality conditions are shown to associate them to certain subsets of
imaginary roots. These methods also allow for an interpretation of Seiberg
duality as the action of the affine Weyl group on the root lattice.
