Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality
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| Format: | Preprint |
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2020
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| author | Douglas, Daniel C. Sun, Zhe |
| author_facet | Douglas, Daniel C. Sun, Zhe |
| contents | 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 $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_14202 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality Douglas, Daniel C. Sun, Zhe Geometric Topology Combinatorics Quantum Algebra Representation Theory 2020: 32G15, 57M15 (Primary), 13F60, 14J33 (Secondary) 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 $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$. |
| title | Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality |
| topic | Geometric Topology Combinatorics Quantum Algebra Representation Theory 2020: 32G15, 57M15 (Primary), 13F60, 14J33 (Secondary) |
| url | https://arxiv.org/abs/2012.14202 |