部分$F$-不变量与簇范畴化
Partial $F$-invariants and cluster categorifications
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中文总结 AI 辅助
本文研究簇代数中部分$F$-不变量的突变公式，证明Reading猜想，并揭示其与部分$E$-不变量及量子仿射代数极点阶数的联系，给出组合公式。
中文摘要 AI 辅助
簇代数中的$F$-不变量是一种组合不变量，它统一了来自加法范畴化的$E$-不变量和来自幺半范畴化的$\mathfrak{d}$-不变量。在本文中，我们研究其细化版本——部分$F$-不变量，并建立了其在初始种子变化下的突变公式。作为应用，我们证明了Reading的一个猜想，该猜想断言：在改变初始种子时，不兼容的簇变量可以通过$g$-向量的符号相干性来区分。我们进一步证明，对于簇单项式，部分$F$-不变量既与带势箭图的可达装饰表示的部分$E$-不变量一致，也与量子仿射代数上有限维可达单模的归一化$R$-矩阵的极点阶数（或部分$\mathfrak{d}$-不变量）一致。作为推论，我们得到了可达单模极点阶数关于$q$-字符的组合公式；我们验证了Kirillov--Reshetikhin模之间极点阶数的猜想显式公式。
英文摘要
The $F$-invariant in cluster algebras is a combinatorial invariant that unifies the $E$-invariant from additive categorification and the $\mathfrak{d}$-invariant from monoidal categorification. In this paper, we study its refinement, the partial $F$-invariant, and establish its mutation formula under changes of the initial seed. As an application, we prove a conjecture of Reading, which asserts that the non-compatible cluster variables can be separated by sign-coherence of $g$-vectors upon varying the initial seed. We further show that, for cluster monomials, the partial $F$-invariants coincide with both the partial $E$-invariants for reachable decorated representations of quivers with potentials and the pole orders of normalized $R$-matrices (or partial $\mathfrak{d}$-invariants) for finite-dimensional reachable simple modules over quantum affine algebras. As consequences, we obtain a combinatorial formula for the pole orders for reachable simple modules in terms of $q$-characters; we verify the conjectural explicit formula for the pole orders between Kirillov--Reshetikhin modules.
发表机构
- University of Science and Technology of China（中国科学技术大学）
- Research Institute for Mathematical Sciences, Kyoto University（京都大学数理解析研究所）
- Kyoto University Institute for Advanced Study (KUIAS), Kyoto University（京都大学高等研究院）
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