Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

TL;DR

This paper establishes a direct link between Dorey's geometric rule and the structure of q-characters in simply-laced quantum affine algebras, providing a criterion for monomial appearance in tensor products.

Contribution

It proves that Dorey's rule precisely characterizes when the monomial 1 appears in the q-character of tensor products, independent of ADE classification.

Findings

Dorey's rule is necessary and sufficient for monomial 1 in q-characters.

The result applies to fundamental representations of simply-laced quantum affine algebras.

The proof does not rely on ADE classification.

Abstract

Let Uq(ghat) be the quantum affine algebra associated to a simply-laced simple Lie algebra g. We examine the relationship between Dorey's rule, which is a geometrical statement about Coxeter orbits of g-weights, and the structure of q-characters of fundamental representations V_{i,a} of Uq(ghat). In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.