Combinatorics of the q-characters of Hernandez-Leclerc modules

Let g be a complex simple Lie algebra and U-q((g) over cap) be the corresponding untwisted quantum affine algebra. In this paper, we give a path description of the q-characters of Hernandez-Leclerc modules, show that up to spectral parameter shift, the equivalent classes of Hernandez-Leclerc modules in the Grothendieck ring of the category of finite-dimensional U-q((sl(n+1)) over cap)-modules are cluster variables in the cluster algebra introduced by Hernandez and Leclerc, and finally prove that the geometric q-character formula conjecture is true for Hernandez-Leclerc modules. (c) 2022 Elsevier Inc. All rights reserved.