  The Demazure character formula is applied to the Verlinde formula for affine
fusion rules. We follow Littelmann's derivation of a generalized
Littlewood-Richardson rule from Demazure characters. A combinatorial rule for
affine fusions does not result, however. Only a modified version of the
Littlewood-Richardson rule is obtained that computes an (old) upper bound on
the fusion coefficients of affine $A_r$ algebras. We argue that this is because
the characters of simple Lie algebras appear in this treatment, instead of the
corresponding affine characters. The Bruhat order on the affine Weyl group must
be implicated in any combinatorial rule for affine fusions; the Bruhat order on
subgroups of this group (such as the finite Weyl group) does not suffice.

High Energy Physics - Theory;Quantum Algebra;Quantum Algebra