Intersection Pairings for Higher Laminations

TL;DR

This paper provides a geometric interpretation of intersection pairings for higher laminations in the case of G=SL_n, relating them to minimal weighted networks in affine buildings, and proves several conjectures from prior work.

Contribution

It introduces a geometric interpretation of intersection pairings as minimal weighted networks in affine buildings for G=SL_n, confirming several conjectures from previous research.

Findings

Intersection pairings can be computed as minimal weighted networks.

The approach relates intersection pairings to the metric structure of affine buildings.

Several conjectures from [LO] are proven using combinatorial tools.

Abstract

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov 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 the case that $G=SL_n$. In particular, we show that they can be computed as the length of minimal weighted networks in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [LO] The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, Konig's theorem, and the Kuhn-Munkres algorithm.