# ALGEBRAIC COMBINATORICS

Intersection Pairings for Higher Laminations
Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 823-841.

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.

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Accepted:
Published online:
DOI: 10.5802/alco.182
Classification: 05B35,  20E42,  90C24,  13F60
Keywords: Discrete geometry, buildings, matroid, convexity, tropical geometry, cluster algebras.
Le, Ian 1

1 Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Le, Ian. Intersection Pairings for Higher Laminations. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 823-841. doi : 10.5802/alco.182. https://alco.centre-mersenne.org/articles/10.5802/alco.182/

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