The positive orthogonal Grassmannian

The Plücker positive region ${\mathrm{O}\mathrm{G}\mathrm{r}}_{+}(k,2k)$ 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 ${\mathrm{O}\mathrm{G}\mathrm{r}}_{+}(k,n)$ for general values of $k,n$. We determine the boundary structure of the quadric ${\mathrm{O}\mathrm{G}\mathrm{r}}_{+}(1,n)$ in ${\mathbb{P}}_{+}^{n-1}$ and show that it is a positive geometry. We show that ${\mathrm{O}\mathrm{G}\mathrm{r}}_{+}(k,2k+1)$ is isomorphic to ${\mathrm{O}\mathrm{G}\mathrm{r}}_{+}(k+1,2k+2)$ 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 ${\mathrm{G}\mathrm{r}}_{+}(k,n)$ do not induce a CW cell decomposition of ${\mathrm{O}\mathrm{G}\mathrm{r}}_{+}(k,n)$.