Intersection Pairings for Higher Laminations

Ian Le

doi:10.5802/alco.182

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.

Highlights

Higher laminations arise as the tropical points of configuration spaces of flags
In [7], we showed that higher laminations on a disc are given by configurations of points in the affine building
We will later define the affine building, but for the only relevant fact is that it is a metric space that behaves in many ways like a symmetric space

Summary

Introduction

The first goal is to give geometric interpretations of the intersection pairings between higher laminations. We show that the intersection pairing is given by the minimal weighted length of a network in the building This suggests that such a geometric interpretation should exist in general, and we explain some conjectures about this at the end of the paper. We expect that conjectural generalizations of Theorem 4.3 discussed in Section 6 will be linearizations of combinatorial statements of minmax type.

Hall’s theorem and its generalizations

Linearization

Main Theorem

Buildings and Laminations

Note that both sides of the expression are

Generalizations