One can realize higher laminations as positive configurations of points in the affine building [7]. The duality pairings of Fock and Goncharov [1] give pairings between higher laminations for two Langlands dual groups $G$ and ${G}^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminations on a topological surface.

We give a geometric interpretation of these intersection pairings in a wide variety of cases. In particular, we show that they can be computed as the minimal weighted length of a network in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [9]. We also suggest the next steps toward giving geometric interpretations of intersection pairings in general.

The key tools are linearized versions of well-known classical results from combinatorics, like Hall’s marriage lemma, König’s theorem, and the Kuhn–Munkres algorithm, which are interesting in themselves.

Revised:

Accepted:

Published online:

Keywords: Discrete geometry, buildings, matroid, convexity, tropical geometry, cluster algebras.

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@article{ALCO_2021__4_5_823_0, author = {Le, Ian}, title = {Intersection {Pairings} for {Higher} {Laminations}}, journal = {Algebraic Combinatorics}, pages = {823--841}, publisher = {MathOA foundation}, volume = {4}, number = {5}, year = {2021}, doi = {10.5802/alco.182}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.182/} }

Le, Ian. Intersection Pairings for Higher Laminations. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 823-841. doi : 10.5802/alco.182. http://archive.numdam.org/articles/10.5802/alco.182/

[1] Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006) no. 103, pp. 1-211 | DOI | MR | Zbl

[2] Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math., Volume 201 (2008) no. 1, pp. 83-146 | DOI | MR | Zbl

[3] Geometry of canonical bases and mirror symmetry, Invent. Math., Volume 202 (2015) no. 2, pp. 487-633 | DOI | MR | Zbl

[4] Donaldson–Thomas transformations of moduli spaces of G-local systems, Adv. Math., Volume 327 (2018), pp. 225-348 | DOI | MR | Zbl

[5] Canonical bases for cluster algebras, J. Amer. Math. Soc., Volume 31 (2018) no. 2, pp. 497-608 | DOI | MR | Zbl

[6] Hives and the fibres of the convolution morphism, Selecta Math. (N.S.), Volume 13 (2007) no. 3, pp. 483-496 | DOI | MR | Zbl

[7] Higher laminations and affine buildings, Geom. Topol., Volume 20 (2016) no. 3, pp. 1673-1735 | DOI | MR | Zbl

[8] Cluster structures on higher Teichmuller spaces for classical groups, Forum Math. Sigma, Volume 7 (2019), e13, 165 pages | DOI | MR | Zbl

[9] Geometry of positive configurations in affine buildings, Doc. Math., Volume 22 (2017), pp. 1519-1538 | DOI | MR | Zbl

[10] Concerning Hall’s theorem, Mathematics in St. Petersburg (Amer. Math. Soc. Transl. Ser. 2), Volume 174, Amer. Math. Soc., Providence, RI, 1996, pp. 73-77 | DOI | MR | Zbl

[11] Discrete convex analysis, Math. Programming, Volume 83 (1998) no. 3, Ser. A, pp. 313-371 | DOI | MR | Zbl

[12] A theorem on independence relations, Quart. J. Math. Oxford Ser., Volume 13 (1942), pp. 83-89 | DOI | MR | Zbl

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