Totally nonnegative critical varieties

TL;DR

This paper investigates the structure of totally nonnegative critical varieties in the Grassmannian, revealing their geometric realization as affine poset cyclohedra and establishing homeomorphism to the second hypersimplex.

Contribution

It introduces a novel geometric description of totally nonnegative critical varieties as images of affine poset cyclohedra and characterizes their boundary stratification.

Findings

Crit^{≥0}_f is the image of an affine poset cyclohedron.

Boundary stratification of Crit^{≥0}_f is defined via a continuous map.

Crit^{≥0}_{k,n} is homeomorphic to the second hypersimplex Δ_{2,n}.

Abstract

We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety Crit$^{\ge0}_f$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of Crit$^{\ge0}_f$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety Crit$^{\ge0}_{k,n}$ is homeomorphic to the second hypersimplex $\Delta_{2,n}$.