Demazure characters and WZW fusion rules

Abstract

The Demazure character formula is applied to the Verlinde formula for the fusion rules of Wess–Zumino–Witten (WZW) models. We follow Littelmann’s derivation of a generalized Littlewood–Richardson rule that computes tensor products for simple Lie algebras. A combinatorial rule for WZW 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 Ar WZW models. 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 WZW fusions; the Bruhat order on subgroups of this group (such as the finite Weyl group) does not suffice.