Classification of real modules in monoidal categorifications of cluster algebras

TL;DR

This paper introduces a conjectural formula for highest $\,\ell$-weight monomials of real modules in quantum affine algebras, verifies it under certain conditions, and generalizes key modules and relations in monoidal categorifications of cluster algebras.

Contribution

It proposes a new conjectural formula for real modules, introduces cluster modules, and generalizes classical relations in the context of monoidal categorifications.

Findings

Verified the conjecture under a reachability condition.

Introduced and characterized cluster modules as reachable real modules.

Derived generalized T-system and exchange relations for $q$-characters.

Abstract

In this paper, we propose a conjectural formula for the highest $\ell$-weight monomial of an arbitrary real module over a simply-laced quantum affine algebra. We verify the conjecture under a multiplicative reachability condition, answering the Hernandez--Leclerc classification problem in monoidal categorifications of cluster algebras under this condition. Moreover, we introduce the notion of cluster modules, generalizing Kirillov--Reshetikhin modules and Hernandez--Leclerc modules as special cases. We prove that cluster modules are reachable real modules, and obtain a system of equations governing $q$-characters of the prime cluster modules, providing a natural generalization of both the classical T-system relations for Kirillov--Reshetikhin modules and the exchange relations for Hernandez--Leclerc modules.