The Kirillov-Reshetikhin conjecture and solutions of T-systems

TL;DR

This paper proves the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras, establishing that their modules' characters solve the Q-system and T-system, with explicit formulas and algebraic proofs applicable to all cases.

Contribution

It provides a purely algebraic proof of the conjecture, extending results to non simply-laced cases and offering explicit character formulas for tensor products.

Findings

Characters of Kirillov-Reshetikhin modules solve the Q-system.

Q-characters satisfy the T-system as functional relations.

T-system can be expressed as an exact sequence.

Abstract

We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras : we prove that the character of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor products. In the proof we show that the Kirillov-Reshetikhin modules are special in the sense of monomials and that their q-characters solve the T-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima with geometric arguments which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.