Xiao-Juan An | Searcharxiv

arXiv2026

Monoidal categorification of generalized cluster algebras and conjectures of Fraser and Gleitz

Hernandez and Leclerc introduced the notion of monoidal categorification of cluster algebras. We define similarly the notion of monoidal categorifications of generalized cluster algebras: an abelian monoidal category $\mathcal M$ is said to be a monoidal categorification of a generalized cluster algebra $\mathcal A$ if the Grothendieck ring of $\mathcal M$ is isomorphic to the upper generalized cluster algebra $\mathcal A^{\mathrm{up}}$, and if cluster monomials (resp. cluster variables) of $\mathcal A$ correspond to classes of real simple (resp. real prime simple) objects of $\mathcal M$. Let $\varepsilon$ be a root of unity such that $\varepsilon^{2\ell}=1$ for some $\ell\in\mathbb{Z}_{\geq 2}$. Denote by $\mathcal{C}_{\varepsilon}$ the category of finite-dimensional modules of the restricted quantum loop algebra $U_\varepsilon^{\res}(L\mathfrak{sl}_k)$ at root $\varepsilon$ of unity, and let $\mathcal{C}_{\varepsilon, ξ}$ be a full subcategory of $\mathcal{C}_{\varepsilon}$ determined by a bipartition $ξ: I \to \{0,1\}$ of the Dynkin diagram. For $k=3$, Gleitz conjectured that the Grothendieck ring of $\mathcal C_{\varepsilon,ξ}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$, and that generalized cluster monomials correspond to classes of simple modules. This conjecture is a special case of a more general conjecture of Fraser. In this paper, we prove the first part of Gleitz's conjecture. More precisely, for $k=3$ and arbitrary $\ell\ge2$, we prove that the Grothendieck ring of $\mathcal C_{\varepsilon,ξ}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$. We also classify the real Kirillov--Reshetikhin modules of $U^{\mathrm{res}}_\varepsilon(L\mathfrak{sl}_3)$ and obtain mutation sequences for the real Kirillov--Reshetikhin modules from the initial seed of the generalized cluster algebra.

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arXiv2025

A path description for $\varepsilon$-characters of representations of type $A$ restricted quantum loop algebras at roots of unity

Fix $\varepsilon^{2\ell}=1$ with $\ell \geq 2$. In this paper, we show that all finite-dimensional simple modules of any restricted quantum loop algebra $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ in a certain category can be transformed into snake modules. We obtain an effective and concrete path description for $\varepsilon$-characters of any simple module with highest $l$-weight of degree two and any Kirillov-Reshetikhin module of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$. As an application of our path description, we obtain a necessary and sufficient condition for the tensor product of two fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. Additionally, we obtain a necessary condition for the tensor product of two or more fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible.

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