Totally Nonnegative Critical Varieties

Abstract

Abstract We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety $\operatorname{Crit}^{\geqslant 0}_f$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of $\operatorname{Crit}^{\geqslant 0}_f$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety $\operatorname{Crit}^{\geqslant 0}_{k,n}$ is homeomorphic to the second hypersimplex $\Delta _{2,n}$.