Tensor products and $q$-characters of HL-modules and monoidal categorifications

TL;DR

This paper classifies prime representations in certain monoidal subcategories of quantum affine algebra modules, characterizes when tensor products are irreducible, and establishes these subcategories as monoidal categorifications of type A cluster algebras.

Contribution

It provides a complete classification of prime representations and criteria for irreducibility of tensor products in these subcategories, linking them to cluster algebra categorification.

Findings

Classified prime representations in the subcategories.

Established necessary and sufficient conditions for tensor product irreducibility.

Proved the subcategories serve as monoidal categorifications of type A cluster algebras.

Abstract

We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type $A$. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type $A$ with coefficients.