Let $\mathcal{C}$ be the category of finite‐dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra $\mathfrak{g}$. Let ${\mathcal{C}}^{-}$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$‐character formula conjectured by Hernandez and Leclerc in types $\mathbb{A}$ and $\mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$‐polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square‐free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category ${\mathcal{C}}_1$ are factorial for Dynkin types $\mathbb{A},\mathbb{D},\mathbb{E}$.

中文翻译：

---

蛇形模块的几何q字符公式

让 $\mathcal{C}$ 是量子仿射代数上有限维模块的类别 ${ü}_q（\widehat{\mathfrak{G}}）$ 简单复李代数 $\mathfrak{G}$。让${\mathcal{C}}^{-}$是Hernandez和Leclerc引入的子类别。我们证明了几何$q$Hernandez和Leclerc猜想的字符公式 $\mathbb{一种}$ 和 $\mathbb{乙}$由Mukhin和Young提出的一类称为蛇模块的简单模块。此外，我们为$F$-与蛇模块关联的通用内核的多项式。作为一个应用程序，我们展示了蛇形模块对应于具有无平方分母的簇单项式，并且我们展示了蛇形模块是真实的模块。我们还表明类别的聚类代数${\mathcal{C}}_{\mathrm{1个}}$ 是Dynkin类型的阶乘 $\mathbb{一种}，\mathbb{d}，\mathbb{Ë}$。