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:

Classification: 05B35, 20E42, 90C24, 13F60

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

@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 = {https://alco.centre-mersenne.org/articles/10.5802/alco.182/} }

TY - JOUR AU - Le, Ian TI - Intersection Pairings for Higher Laminations JO - Algebraic Combinatorics PY - 2021 DA - 2021/// SP - 823 EP - 841 VL - 4 IS - 5 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.182/ UR - https://doi.org/10.5802/alco.182 DO - 10.5802/alco.182 LA - en ID - ALCO_2021__4_5_823_0 ER -

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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