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 $\mathcal{A}_{\mathrm{PGL}_3, \widehat{S}}^+(\mathbb{Z}^t)$.
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Keywords: Webs, tropical coordinates, cluster transformations
Douglas, Daniel C. 1; Sun, Zhe 2

@article{ALCO_2025__8_1_101_0, author = {Douglas, Daniel C. and Sun, Zhe}, title = {Tropical {Fock{\textendash}Goncharov} coordinates for $\mathrm{SL}_3$-webs on surfaces {II:} naturality}, journal = {Algebraic Combinatorics}, pages = {101--156}, publisher = {The Combinatorics Consortium}, volume = {8}, number = {1}, year = {2025}, doi = {10.5802/alco.408}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.408/} }
TY - JOUR AU - Douglas, Daniel C. AU - Sun, Zhe TI - Tropical Fock–Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality JO - Algebraic Combinatorics PY - 2025 SP - 101 EP - 156 VL - 8 IS - 1 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.408/ DO - 10.5802/alco.408 LA - en ID - ALCO_2025__8_1_101_0 ER -
%0 Journal Article %A Douglas, Daniel C. %A Sun, Zhe %T Tropical Fock–Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality %J Algebraic Combinatorics %D 2025 %P 101-156 %V 8 %N 1 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.408/ %R 10.5802/alco.408 %G en %F ALCO_2025__8_1_101_0
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/
[1] The localized skein algebra is Frobenius, Algebr. Geom. Topol., Volume 17 (2017) no. 6, pp. 3341-3373 | DOI | MR | Zbl
[2] Non-elliptic webs and convex sets in the affine building, Doc. Math., Volume 25 (2020), pp. 2413-2443 | DOI | MR | Zbl
[3] A duality map for quantum cluster varieties from surfaces, Adv. Math., Volume 306 (2017), pp. 1164-1208 | DOI | MR
[4] Rings of -characters and the Kauffman bracket skein module, Comment. Math. Helv., Volume 72 (1997) no. 4, pp. 521-542 | DOI | MR
[5] Webs and quantum skew Howe duality, Math. Ann., Volume 360 (2014) no. 1-2, pp. 351-390 | DOI | MR
[6] Points of quantum coming from quantum snakes, Algebr. Geom. Topol., Volume 24 (2024), pp. 2537-2570 | DOI | MR
[7] Quantum traces for : the case , J. Pure Appl. Algebra, Volume 228 (2024), Paper no. 107652, 50 pages | DOI | MR | Zbl
[8] Tropical Fock-Goncharov coordinates for -webs on surfaces II: naturality (2020) | arXiv
[9] Tropical Fock-Goncharov coordinates for -webs on surfaces I: construction, Forum Math. Sigma, Volume 12 (2024), Paper no. e5, 55 pages | DOI | MR | Zbl
[10] Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci., Volume 103 (2006), pp. 1-211 | DOI | Numdam | MR | Zbl
[11] Dual Teichmüller and lamination spaces, Handbook of Teichmüller theory. Vol. I (IRMA Lect. Math. Theor. Phys.), Volume 11, Eur. Math. Soc., Zürich, 2007, pp. 647-684 | DOI | MR | Zbl
[12] Moduli spaces of convex projective structures on surfaces, Adv. Math., Volume 208 (2007) no. 1, pp. 249-273 | DOI | MR | Zbl
[13] Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 6, pp. 865-930 | DOI | Numdam | MR
[14] Introduction to cluster algebras. Chapters 4–5 (2017) | arXiv
[15] Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl
[16] Buildings, spiders, and geometric Satake, Compos. Math., Volume 149 (2013) no. 11, pp. 1871-1912 | DOI | MR | Zbl
[17] -skein algebras and webs on surfaces, Math. Z., Volume 300 (2022) no. 1, pp. 33-56 | DOI | MR | Zbl
[18] Spectral networks, Ann. Henri Poincaré, Volume 14 (2013) no. 7, pp. 1643-1731 | DOI | MR | Zbl
[19] Geometry of canonical bases and mirror symmetry, Invent. Math., Volume 202 (2015) no. 2, pp. 487-633 | DOI | MR | Zbl
[20] Donaldson-Thomas transformations of moduli spaces of G-local systems, Adv. Math., Volume 327 (2018), pp. 225-348 | DOI | MR | Zbl
[21] Quantum geometry of moduli spaces of local systems and representation theory (2019) | arXiv
[22] Canonical bases for cluster algebras, J. Amer. Math. Soc., Volume 31 (2018) no. 2, pp. 497-608 | DOI | MR | Zbl
[23] Ueber die Theorie der algebraischen Formen, Math. Ann., Volume 36 (1890) no. 4, pp. 473-534 | DOI | MR | Zbl
[24] McShane identities for higher Teichmüller theory and the Goncharov–Shen potential, Mem. Amer. Math. Soc., Volume 286 (2023), p. v+116 | DOI | MR | Zbl
[25] Unbounded -laminations and their shear coordinates (2022) | arXiv
[26] Quantum decorated character stacks (2021) | arXiv
[27] -laminations as bases for cluster varieties for surfaces (2020) | arXiv
[28] Naturality of quantum trace maps for surfaces (2021) | arXiv
[29] The honeycomb model of tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., Volume 12 (1999) no. 4, pp. 1055-1090 | DOI | MR
[30] Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008) | arXiv
[31] Spiders for rank Lie algebras, Comm. Math. Phys., Volume 180 (1996) no. 1, pp. 109-151 | DOI | MR | Zbl
[32] Higher laminations and affine buildings, Geom. Topol., Volume 20 (2016) no. 3, pp. 1673-1735 | DOI | MR | Zbl
[33] An approach to higher Teichmüller spaces for general groups, Int. Math. Res. Not. IMRN, Volume 2019 (2019), pp. 4899-4949 | DOI | MR
[34] Cluster structures on higher Teichmuller spaces for classical groups, Forum Math. Sigma, Volume 7 (2019), Paper no. e13, 165 pages | DOI | MR
[35] Intersection pairings for higher laminations, Algebr. Comb., Volume 4 (2021) no. 5, pp. 823-841 | DOI | Numdam | MR
[36] Introduction to tropical geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, Providence, RI, 2015, xii+363 pages | DOI | MR
[37] Skein and cluster algebras of marked surfaces, Quantum Topol., Volume 7 (2016) no. 3, pp. 435-503 | DOI | MR
[38] Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag, Berlin, 1994, xiv+292 pages | DOI | MR
[39] The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys., Volume 113 (1987) no. 2, pp. 299-339 | DOI | MR
[40] Skein modules of -manifolds, Bull. Polish Acad. Sci. Math., Volume 39 (1991) no. 1-2, pp. 91-100 | MR | Zbl
[41] On skein algebras and -character varieties, Topology, Volume 39 (2000) no. 1, pp. 115-148 | DOI | MR | Zbl
[42] On total dual integrality, Linear Algebra Appl., Volume 38 (1981), pp. 27-32 | DOI | MR | Zbl
[43] Intersections of Dual -Webs (2023) | arXiv
[44] -character varieties as spaces of graphs, Trans. Amer. Math. Soc., Volume 353 (2001) no. 7, pp. 2773-2804 | DOI | MR | Zbl
[45] Skein theory for -quantum invariants, Algebr. Geom. Topol., Volume 5 (2005), pp. 865-897 | DOI | MR | Zbl
[46] Confluence theory for graphs, Algebr. Geom. Topol., Volume 7 (2007), pp. 439-478 | DOI | MR | Zbl
[47] Flows on the -Hitchin component, Geom. Funct. Anal., Volume 30 (2020) no. 2, pp. 588-692 | DOI | MR | Zbl
[48] Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997, x+311 pages | DOI | MR
[49] Algebras of loops on surfaces, algebras of knots, and quantization, Braid group, knot theory and statistical mechanics (Adv. Ser. Math. Phys.), Volume 9, World Sci. Publ., Teaneck, NJ, 1989, pp. 59-95 | DOI | MR | Zbl
[50] An invitation to higher Teichmüller theory, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ (2018), pp. 1013-1039 | MR | Zbl
[51] Quantum field theory and the Jones polynomial, Comm. Math. Phys., Volume 121 (1989) no. 3, pp. 351-399 | DOI | MR | Zbl
[52] Higher laminations, webs and N=2 line operators (2013) | arXiv
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