Intersections of dual 𝑆𝐿₃-webs

Shen, Linhui; Sun, Zhe; Weng, Daping

doi:10.1090/tran/9349

Intersections of dual $\operatorname {SL}_3$-webs
HTML articles powered by AMS MathViewer

by Linhui Shen, Zhe Sun and Daping Weng;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9349

Published electronically: May 1, 2025

HTML | PDF | Request permission

Abstract:

We introduce a topological intersection number for an ordered pair of $\operatorname {SL}_3$-webs on a decorated surface. Using this intersection pairing between reduced $(\operatorname {SL}_3,\mathcal {A})$-webs and a collection of $(\operatorname {SL}_3,\mathcal {X})$-webs associated with the Fock–Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced $(\operatorname {SL}_3,\mathcal {A})$-webs to the tropical set $\mathcal {A}^+_{\operatorname {PGL}_3,\hat {S}}(\mathbb {Z}^t)$, as established by Douglas and Sun in [Forum Math. Sigma 12 (2024), p. e5, 55]. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fock–Goncharov duality conjecture of higher Teichmüller spaces for $\operatorname {SL}_3$.

References

Tair Akhmejanov, Non-elliptic webs and convex sets in the affine building, Doc. Math. 25 (2020), 2413–2443. MR 4213127
Francis Bonahon and Helen Wong, Quantum traces for representations of surface groups in $\textrm {SL}_2(\Bbb C)$, Geom. Topol. 15 (2011), no. 3, 1569–1615. MR 2851072, DOI 10.2140/gt.2011.15.1569
Daniel Douglas and Zhe Sun, Tropical fock-goncharov coordinates for sl3-webs on surfaces II: naturality, , Algebr. Comb. 8 (2025), no. 1, 101–156.
Daniel C. Douglas and Zhe Sun, Tropical Fock-Goncharov coordinates for $\textrm {SL}_3$-webs on surfaces I: construction, Forum Math. Sigma 12 (2024), Paper No. e5, 55. MR 4685684, DOI 10.1017/fms.2023.120
Jiarui Fei, Tropical $F$-polynomials and general presentations, J. Lond. Math. Soc. (2) 107 (2023), no. 6, 2079–2120. MR 4598180, DOI 10.1112/jlms.12734
Vladimir Fock and Alexander Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211. MR 2233852, DOI 10.1007/s10240-006-0039-4
V. V. Fock and A. B. Goncharov, Cluster Poisson varieties at infinity, Selecta Math. (N.S.) 22 (2016), no. 4, 2569–2589. MR 3573965, DOI 10.1007/s00029-016-0282-6
Vladimir V. Fock and Alexander B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865–930 (English, with English and French summaries). MR 2567745, DOI 10.24033/asens.2112
Sergey Fomin and Pavlo Pylyavskyy, Tensor diagrams and cluster algebras, Adv. Math. 300 (2016), 717–787. MR 3534844, DOI 10.1016/j.aim.2016.03.030
Bruce Fontaine, Generating basis webs for $\textrm {SL_n}$, Adv. Math. 229 (2012), no. 5, 2792–2817. MR 2889146, DOI 10.1016/j.aim.2012.01.016
Bruce Fontaine, Joel Kamnitzer, and Greg Kuperberg, Buildings, spiders, and geometric Satake, Compos. Math. 149 (2013), no. 11, 1871–1912. MR 3133297, DOI 10.1112/S0010437X13007136
Chris Fraser, Webs and canonical bases in degree two, Comb. Theory 3 (2023), no. 3, Paper No. 11, 26. MR 4683618
Chris Fraser and Pavlo Pylyavskyy, Tensor diagrams and cluster combinatorics at punctures, Adv. Math. 412 (2023), Paper No. 108796, 83. MR 4520422, DOI 10.1016/j.aim.2022.108796
Charles Frohman and Adam S. Sikora, $SU(3)$-skein algebras and webs on surfaces, Math. Z. 300 (2022), no. 1, 33–56. MR 4359515, DOI 10.1007/s00209-021-02765-z
Alexander Goncharov and Linhui Shen, Geometry of canonical bases and mirror symmetry, Invent. Math. 202 (2015), no. 2, 487–633. MR 3418241, DOI 10.1007/s00222-014-0568-2
Alexander Goncharov and Linhui Shen, Donaldson-Thomas transformations of moduli spaces of G-local systems, Adv. Math. 327 (2018), 225–348. MR 3761995, DOI 10.1016/j.aim.2017.06.017
Alexander Goncharov and Linhui Shen, Quantum geometry of moduli spaces of local systems and representation theory, arXiv:1904.10491 Preprint, 2019.
Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 497–608. MR 3758151, DOI 10.1090/jams/890
Yi Huang and Zhe Sun, McShane identities for higher Teichmüller theory and the Goncharov-Shen potential, Mem. Amer. Math. Soc. 286 (2023), no. 1422, v+116. MR 4595283, DOI 10.1090/memo/1422
Tsukasa Ishibashi and Shunsuke Kano, Unbounded $\operatorname {sl}_3$-laminations and their shear coordinates, Preprint, arXiv:2204.08947 2022.
Tsukasa Ishibashi, Hironori Oya, and Linhui Shen, $\mathcal {A}=\mathcal {U}$ for cluster algebras from moduli spaces of $G$-local systems, Adv. Math. 431 (2023), Paper No. 109256, 50. MR 4631996, DOI 10.1016/j.aim.2023.109256
Tsukasa Ishibashi, Zhe Sun, and Wataru Yuasa, Bounded $\mathfrak {sp}_4$-laminations and their intersection coordinates, In preparation.
Tsukasa Ishibashi and Wataru Yuasa, Skein and cluster algebras of unpunctured surfaces for $\mathfrak {sp}_4$, Adv. Math. 465 (2025), Paper No. 110149.
Tsukasa Ishibashi and Wataru Yuasa, Skein and cluster algebras of unpunctured surfaces for $\mathfrak {sl}_3$, Math. Z. 303 (2023), no. 3, Paper No. 72, 60. MR 4552137, DOI 10.1007/s00209-023-03208-7
Hyun Kyu Kim, $\mathbf {SL}_3$-laminations as bases for $\mathbf {PGL}_3$ cluster varieties for surfaces, Mem. Amer. Math. Soc. (2020), To appear.
Allen Knutson and Terence Tao, The honeycomb model of $\textrm {GL}_n(\textbf {C})$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055–1090. MR 1671451, DOI 10.1090/S0894-0347-99-00299-4
Allen Knutson, Terence Tao, and Christopher Woodward, A positive proof of the Littlewood-Richardson rule using the octahedron recurrence, Electron. J. Combin. 11 (2004), no. 1, Research Paper 61, 18. MR 2097327, DOI 10.37236/1814
Greg Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, 109–151. MR 1403861
Ian Le, Intersection pairings for higher laminations, Algebr. Comb. 4 (2021), no. 5, 823–841. MR 4339354, DOI 10.5802/alco.182
Thang T. Q. Lê and Tao Yu, Quantum traces for $\operatorname {SL}_n$-skein algebras, Preprint, arXiv:2303.08082 2023.
Travis Mandel and Fan Qin, Bracelet bases are theta bases, Preprint, arXiv:2301.11101 2023.
Andrew Neitzke and Fei Yan, The quantum UV-IR map for line defects in $\mathfrak {gl} (3)$-type class $S$ theories, J. High Energy Phys. 9 (2022), Paper No. 81, 50. MR 4482975, DOI 10.1007/jhep09(2022)081
R. C. Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), no. 2, 299–339. MR 919235
Gus Schrader, Linhui Shen, and Eric Zaslow, The chromatic lagrangian: wavefunctions and open gromov-witten conjectures, Adv. Theor. Math. Phys., to appear (2023).
Linhui Shen, Cluster nature of quantum groups, Preprint, arXiv:2209.06258 2022.
Linhui Shen, Zhe Sun, and Daping Weng, The punctured $\operatorname {SL}_3$ skein algebra and the quantization of $\mathcal {A}_{\operatorname {sl}_3,\hat {s}}$ moduli space, In preparation.
Adam S. Sikora and Bruce W. Westbury, Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007), 439–478. MR 2308953, DOI 10.2140/agt.2007.7.439
William P. Thurston, The geometry and topology of three-manifolds. Vol. IV, American Mathematical Society, Providence, RI, [2022] ©2022. Edited and with a preface by Steven P. Kerckhoff and a chapter by J. W. Milnor. MR 4554426

AMS Website Down

Transactions of the American Mathematical Society

Intersections of dual $\operatorname {SL}_3$-webs
HTML articles powered by AMS MathViewer

Abstract: