Fusion bases as facets of polytopes

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 ŝu(2), 432 for ŝu(3) and quite surprisingly, it reduces to 36 for ŝu(4), while it is only of order 4 for ŝp(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 ŝu(2) and ŝu(3), it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For ŝu(3), new symmetries of the fusion coefficients are found. 05/00-08/01 (hepth@xxx/0108213) 1 Work supported by NSERC (Canada). 2 Work supported by NSERC (Canada) and FCAR (Québec).