Lattice path matroidal subdivisions, Positive Tropical Grassmannian and Amplituhedron

TL;DR

This paper introduces lattice path matroidal subdivisions and explores their connection to the positive tropical Grassmannian and amplituhedron, revealing new geometric structures relevant to physics and combinatorics.

Contribution

It defines LPM subdivisions, shows they are regular, and links them to the positive Dressian and tropical Grassmannian, advancing understanding of their geometric and physical significance.

Findings

LPM subdivisions are regular and lie in the Dressian.

The positive Dressian equals the positive tropical Grassmannian.

Connections to the amplituhedron and positive configuration space are established.

Abstract

We introduce the notion of lattice path matroidal subdivisions, or LPM subdivisions for short, and show that these subdivisions are regular and hence the weight vectors for them lie in the Dressian. This leads us to explore the structure of the set of these weights inside the Dressian and owing to the fact that Lattice path matroids are positroids, we move to the positive Dressian which in turn is equal to the positive tropical Grassmannian, an object of immense interest currently in Physics. This is related to the amplituhedron and positive configuration space, which we describe here and wish to explore these connections further.