The positive orthogonal Grassmannian
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915206034096128 |
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| author | Maazouz, Yassine El Mandelshtam, Yelena |
| author_facet | Maazouz, Yassine El Mandelshtam, Yelena |
| contents | The Plücker 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 $\mathrm{OGr}_+(k,n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14091 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The positive orthogonal Grassmannian Maazouz, Yassine El Mandelshtam, Yelena Combinatorics Algebraic Geometry 05E14, 14M15, 15B48 The Plücker 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 $\mathrm{OGr}_+(k,n)$. |
| title | The positive orthogonal Grassmannian |
| topic | Combinatorics Algebraic Geometry 05E14, 14M15, 15B48 |
| url | https://arxiv.org/abs/2412.14091 |