Demazure Characters and Affine Fusion Rules

TL;DR

This paper explores the application of Demazure characters to affine fusion rules, revealing limitations in deriving a combinatorial rule for affine fusions and highlighting the role of the affine Weyl group’s Bruhat order.

Contribution

It demonstrates that Demazure characters yield an upper bound for affine fusion coefficients and emphasizes the importance of the affine Weyl group's Bruhat order in formulating fusion rules.

Findings

Demazure characters provide an upper bound on affine fusion coefficients.

A combinatorial rule for affine fusions remains elusive.

The affine Weyl group's Bruhat order is essential for affine fusion rules.

Abstract

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.