Prime representations in the Hernandez-Leclerc category: classical decompositions
Leon Barth, Deniz Kus
TL;DR
The paper analyzes prime irreducible objects in the Hernandez–Leclerc category $\mathcal{C}_{q,\kappa}$ for the quantum affine algebra of $\mathfrak{sl}_{n+1}$ and their graded limits to the current algebra. By employing the dual functional realization of loop algebras and introducing truncated Weyl modules $M_{\boldsymbol{\xi},\lambda}$, it develops a polyhedral framework to describe the graded decompositions of the graded limits $L(\boldsymbol{\pi})$ for $\boldsymbol{\pi}\in\mathcal{P}_{\mathbb{Z}}^+(1)$. The main result expresses the graded multiplicities as lattice-point counts in explicitly defined convex polytopes, enabling concrete computations and connecting to level-two Demazure modules and the cluster-algebra interpretation of the HL category. Together with prior work, these findings yield graded decompositions of stable prime Demazure modules and reinforce the interplay between quantum affine representations, cluster algebras, and geometric combinatorics via polytope calculus.
Abstract
We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez-Leclerc category for the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. When the HL category is realized as a monoidal categorification of a cluster algebra \cite{HL10,HL13}, these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $\mathbf{U}_q(\mathfrak{sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with \cite{BCMo15} we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.