Fusion algebras were defined by Fuchs, see [8], and a wide class of examples were found in the representation theory of affine Lie algebras, where they appear as a two parameter family  with  the Lie algebra. It was conjectured for many years that the structure constants  for this algebras satisfy the level increasing property which establishes that

 for all l, m and n and for all

Recently this conjecture was proved by Feingold and Fredenhagen, see [4], by using algebraic techniques.

In this paper, we give a different proof of a particular case of this level increasing property for type A fusion algebras by using a pure combinatorial argument which involves orbits of  which were also introduced by Feingold and his co-author Weiner, see [2].

fusionalgebras, fusion coefficients, affine Lie algebra, orbits of