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 . 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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.182
Classification: 05B35,  20E42,  90C24,  13F60
Keywords: Discrete geometry, buildings, matroid, convexity, tropical geometry, cluster algebras.
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     title = {Intersection {Pairings} for {Higher} {Laminations}},
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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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