The positive orthogonal Grassmannian

TL;DR

This paper explores the structure and properties of the positive orthogonal Grassmannian for general parameters, revealing its boundary structure, isomorphisms, and combinatorial connections relevant to positive geometry and scattering amplitudes.

Contribution

It generalizes the study of the positive orthogonal Grassmannian, determines its boundary structure, and uncovers new combinatorial and geometric relationships.

Findings

The boundary structure of $ ext{OGr}_+(1,n)$ is a positive geometry.

$ ext{OGr}_+(k,2k+1)$ is isomorphic to $ ext{OGr}_+(k+1, 2k+2)$.

Positroid cells do not form a CW decomposition of $ ext{OGr}_+(k,n)$ for $n>2k+1$.

Abstract

The Pl\"ucker positive region $\mathrm{OGr}_+(k,2k)$ of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian $\mathrm{OGr}_+(k,n)$ for general values of $k,n$. We determine the boundary structure of the quadric $\mathrm{OGr}_+(1,n)$ in $\mathbb{P}^{n-1}_{+}$ and show that it is a positive geometry. We show that $\mathrm{OGr}_+(k,2k+1)$ is isomorphic to $\mathrm{OGr}_+(k+1, 2k+2)$ and connect its combinatorial structure to matchings on $[2k+2]$. Finally, we show that in the case $n>2k+1$, the \emph{positroid cells} of $\mathrm{Gr}_+(k,n)$ do not induce a CW cell decomposition of $\mathrm{OGr}_+(k,n)$.