论文标题
$ \ mathrm {sl} _3 $ -webs在表面II：自然的热带Fock-Goncharov坐标
Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality
论文作者
论文摘要
在同伴论文（Arxiv 2011.01768）中，我们构建了非负整数协调$φ_\ MATHSCR {t}（\ Mathscr {w} _ {3，\ hat \ hat {s}}}}} $ \ mathscr {w} _ {3，\ hat \ hat {s}} $减少的$ \ m artrm {sl} _3 $ -webs在有限型式刺穿的表面$ \ hat {s} $上，取决于理想的三角形$ \ m artsscr {我们表明，这些坐标在选择三角剖分方面是自然的，从某种意义上说，如果选择了不同的三角剖分$ \ mathscr {t}^\ prime $，则可以选择$φ_\ mathscr {t}（t}（\ mathscr {w} _ {3，\ at {3，s { $φ_ {\ MATHSCR {t}^\ prime}（\ Mathscr {w} _ {3，\ hat {s}}）$是热带$ \ MATHCAL {A} $ - 坐标 - 坐标 - 坐标 - 坐标 - 坐标群集转换。因此，我们可以将网络$ \ mathscr {w} _ {3，\ hat {s}} $作为fock-goncharov-shen积极整数热带点$ \ mathcal {a} _ { \ hat {s}}}^+（\ mathbb {z}^t）$。
In a companion paper (arXiv 2011.01768), we constructed nonnegative integer coordinates $Φ_\mathscr{T}(\mathscr{W}_{3, \hat{S}}) \subset \mathbb{Z}_{\geq 0}^N$ for the collection $\mathscr{W}_{3, \hat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\hat{S}$, depending on an ideal triangulation $\mathscr{T}$ of $\hat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathscr{T}^\prime$ is chosen, then the coordinate change map relating $Φ_\mathscr{T}(\mathscr{W}_{3, \hat{S}})$ to $Φ_{\mathscr{T}^\prime}(\mathscr{W}_{3, \hat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathscr{W}_{3, \hat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$.
