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Tropical Fock–Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality
Douglas, Daniel C.1; Sun, Zhe2
1 Virginia Tech Department of Mathematics 225 Stanger Street Blacksburg VA 24061 (USA)
2 University of Science and Technology of China School of Mathematical Sciences 96 Jinzhai Road Hefei 230026 (China)
Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 101-156.

In a companion article, we constructed nonnegative integer coordinates $\Phi _\mathcal{T}(\mathcal{W}_{3, \widehat{S}}) \subset \mathbb{Z}_{\ge 0}^N$ for the collection $\mathcal{W}_{3, \widehat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\widehat{S}$, depending on an ideal triangulation $\mathcal{T}$ of $\widehat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathcal{T}^\prime $ is chosen, then the coordinate change map relating $\Phi _\mathcal{T}(\mathcal{W}_{3, \widehat{S}})$ to $\Phi _{\mathcal{T}^\prime }(\mathcal{W}_{3, \widehat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathcal{W}_{3, \widehat{S}}$ as a concrete topological model for the Fock–Goncharov–Shen positive integer tropical points $\mathcal{A}_{\mathrm{PGL}_3, \widehat{S}}^+(\mathbb{Z}^t)$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.408
Classification: 32G15, 57M15, 13F60, 14J33
Keywords: Webs, tropical coordinates, cluster transformations

Douglas, Daniel C. 1; Sun, Zhe 2

1 Virginia Tech Department of Mathematics 225 Stanger Street Blacksburg VA 24061 (USA)
2 University of Science and Technology of China School of Mathematical Sciences 96 Jinzhai Road Hefei 230026 (China)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Douglas, Daniel C.; Sun, Zhe. Tropical Fock–Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality. Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 101-156. doi : 10.5802/alco.408. https://alco.centre-mersenne.org/articles/10.5802/alco.408/

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