Positive Grassmannian and polyhedral subdivisions

TL;DR

This paper explores the positive Grassmannian's rich combinatorial structures, linking them to polyhedral subdivisions and hypersimplex projections, revealing new geometric and algebraic insights with broad mathematical and physical implications.

Contribution

It introduces a novel perspective by identifying plabic and Grassmannian graphs with polyhedral subdivisions from hypersimplex projections, connecting the positive Grassmannian to fiber polytopes and the Baues problem.

Findings

Identified plabic graphs with polyhedral subdivisions

Linked positive Grassmannian structures to fiber polytopes

Suggested extensions of Grassmannian-related objects

Abstract

The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial structures appeared in many other areas of mathematics and physics, e.g., in the study of cluster algebras, scattering amplitudes, and solitons. We discuss new ways to think about these structures. In particular, we identify plabic graphs and more general Grassmannian graphs with polyhedral subdivisions induced by 2-dimensional projections of hypersimplices. This implies a close relationship between the positive Grassmannian and the theory of fiber polytopes and the generalized Baues problem. This suggests natural extensions of objects related to the positive Grassmannian.