  A new way of constructing fusion bases (i.e., the set of inequalities
governing fusion rules) out of fusion elementary couplings is presented. It
relies on a polytope reinterpretation of the problem: the elementary couplings
are associated to the vertices of the polytope while the inequalities defining
the fusion basis are the facets. The symmetry group of the polytope associated
to the lowest rank affine Lie algebras is found; it has order 24 for $\su(2)$,
432 for $\su(3)$ and quite surprisingly, it reduces to 36 for $\su(4)$, while
it is only of order 4 for $\sp(4)$. This drastic reduction in the order of the
symmetry group as the algebra gets more complicated is rooted in the presence
of many linear relations between the elementary couplings that break most of
the potential symmetries. For $\su(2)$ and $\su(3)$, it is shown that the
fusion-basis defining inequalities can be generated from few (1 and 2
respectively) elementary ones. For $\su(3)$, new symmetries of the fusion
coefficients are found.
