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The positive orthogonal Grassmannian

Yassine El Maazouz, Yelena Mandelshtam

TL;DR

This work initiates a general study of the positive orthogonal Grassmannian $\mathrm{OGr}_+^{\omega}(k,n)$, investigating its boundary geometry, combinatorics, and positivity structures across all $k,n$. It develops a robust algebraic framework, establishing the defining ideal via Plücker relations and quadratic orthogonality constraints, computing dimensions and degrees (via Weyl dimensions) and proving primality for $n>2k$. It then provides concrete positive-geometry descriptions in the $k=1$ case, including a canonical form and a product-of-simplices combinatorial model, and identifies an isomorphism $\mathrm{OGr}_+(k,2k+1) \cong \mathrm{OGr}_+(k+1,2k+2)$ tied to matchings on $[2k+2]$. Finally, it shows that for $n>2k+1$ the positroid-cell decomposition of $\mathrm{Gr}_+(k,n)$ does not yield a CW decomposition of $\mathrm{OGr}_+^{\omega_0}(k,n)$, motivating the introduction of orthopositroids and the development of new combinatorial tools to describe boundary structures. This advances both the algebraic-geometry and combinatorial understanding of positive orthogonal Grassmannians with potential implications for related physical amplitudes.

Abstract

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 $\mathrm{OGr}_+(k,n)$.

The positive orthogonal Grassmannian

TL;DR

This work initiates a general study of the positive orthogonal Grassmannian , investigating its boundary geometry, combinatorics, and positivity structures across all . It develops a robust algebraic framework, establishing the defining ideal via Plücker relations and quadratic orthogonality constraints, computing dimensions and degrees (via Weyl dimensions) and proving primality for . It then provides concrete positive-geometry descriptions in the case, including a canonical form and a product-of-simplices combinatorial model, and identifies an isomorphism tied to matchings on . Finally, it shows that for the positroid-cell decomposition of does not yield a CW decomposition of , motivating the introduction of orthopositroids and the development of new combinatorial tools to describe boundary structures. This advances both the algebraic-geometry and combinatorial understanding of positive orthogonal Grassmannians with potential implications for related physical amplitudes.

Abstract

The Plücker positive region 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 for general values of . We determine the boundary structure of the quadric in and show that it is a positive geometry. We show that is isomorphic to and connect its combinatorial structure to matchings on . Finally, we show that in the case , the \emph{positroid cells} of do not induce a CW cell decomposition of .

Paper Structure

This paper contains 5 sections, 12 theorems, 69 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Commutative algebra and geometry of
  3. The positive orthogonal Grassmannian
  4. The positive Orthogonal Grassmannian
  5. What goes wrong for when and ?

Key Result

Theorem 2.1

The orthogonal Grassmannian $\mathop{\mathrm{OGr}}\nolimits^\omega(k,n)$ is cut out in $\mathbb{P}^{\binom{n}{k} - 1}$ by the Plücker relations in addition to the following $\frac{1}{2}\binom{n}{k-1} (\binom{n}{k-1}+1)$ equations: where $\epsilon(I\ell) = (-1)^{|\{i \in I \colon i > \ell \}|}$ denotes the sign of the permutation that sorts $I\ell$.

Figures (7)

  • Figure 1: The poset $\mathcal{P}_{2,6}$ is created from $Y_{2,6}$ and $\widetilde{Y}_{2,6}$ by adding the six covering relations in red.
  • Figure 2: A lattice path depiction of the bijection in Lemma \ref{['lem:CountPairs']}. The red path crosses the reflection of the blue path, pictured as a dotted line.
  • Figure 3: The Hasse diagram of the poset structure on $\mathfrak{S}_{1,5}$.
  • Figure 4: The positive Grassmannian $\mathop{\mathrm{OGr}}\nolimits_{+}{(1,4)}$ is the red region in the tetrahedron $\mathbb{P}^3_{+}$. The boundaries of $\mathop{\mathrm{OGr}}\nolimits_{+}(1,4)$ lie on the facets of $\mathbb{P}^3_{+}$.
  • Figure 5: The face poset of $\mathop{\mathrm{OGr}}\nolimits_+(2,5)$ matches that of $\mathop{\mathrm{OGr}}\nolimits_+(3,6)$. See Figure 7 in galashin2020ising.
  • ...and 2 more figures

Theorems & Definitions (39)

  • Definition 1.1
  • Example 1.2
  • Theorem 2.1
  • proof
  • Example 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Theorem 2.6
  • ...and 29 more