辫子簇的编织上的突变序列与融合
Mutation Sequences along Weaves and Amalgamation of Braid Varieties
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中文总结 AI 辅助
本文通过沿双Demazure编织的突变序列构造簇局部化，证明双Bott-Samelson胞腔端点层的分解为拟簇同构，并验证半装饰情形下Gorsky-Kim-Scroggin-Simental猜想。
中文摘要 AI 辅助
设 $p,q$ 为具有 Demazure 积 $u,v$ 的正辫子。双 Bott-Samelson 胞腔的端点层 $\mathrm{Conf}(p,q)_{u,v}$ 分解为 $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$，即一个双 Bruhat 胞腔乘以两个辫子簇。我们证明该层的簇结构由 $\mathrm{Conf}(p,q)$ 上的簇结构的簇局部化给出，且分解映射是拟簇同构。这源于沿双 Demazure 编织的突变序列，每个三价顶点对应一个突变，终止于双词箭图与编织箭图的扩展的融合。在半装饰情形下，这证明了 Gorsky-Kim-Scroggin-Simental 关于拼接映射 $X(p) \times X(\Delta\Delta) \to X(p\Delta)$ 的猜想，其中 $\Delta$ 是 $w_0$ 的约化正辫子且 $\mathrm{dem}(p) = w_0$。
英文摘要
Let $p,q$ be positive braids with Demazure products $u,v$. The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on $\mathrm{Conf}(p,q)$, and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map $X(p) \times X(ΔΔ) \to X(pΔ)$, where $Δ$ is a reduced positive braid for $w_0$ and $\mathrm{dem}(p) = w_0$.
