[Paper Review] An explicit description in terms of Plücker coordinates of the Langrangian-Grassmannian

Jesús Carrillo Pacheco, Fausto Jarquín-Zárate+2 more|arXiv (Cornell University)|Jan 27, 2016

Coding theory and cryptography3 references3 citations

TL;DR

This paper provides an explicit, characteristic-free description of the Lagrangian-Grassmannian $L(n,2n)$ as a linear section of the Grassmannian $G(n,2n)$ in Plücker coordinates. Using a symplectic basis and contraction maps, it derives explicit linear equations—indexed by certain multi-indices—that cut out the kernel of the contraction map, thereby characterizing the Lagrangian-Grassmannian as the common zero locus of Plücker quadratic relations and these linear forms, valid over any field.

ABSTRACT

For an arbitrary field of any characteristic we give an explicit description, in terms of Plücker coordinates, of the projective linear space that cuts out the Lagrangian-Grassmannian variety $L(n,2n)$ of maximal isotropic subspaces in a symplectic vector space of dimension $2n$ in the Grassmannian variety $G(n,2n)$

Motivation & Objective

• To provide an explicit, coordinate-based characterization of the Lagrangian-Grassmannian $L(n,2n)$ in terms of Plücker coordinates over an arbitrary field.
• To determine the linear equations that define the kernel of the contraction map $f: igwedge^n E o igwedge^{n-2} E$ induced by the symplectic form.
• To give a uniform, explicit description of the linear section $\mathbb{P}(\ker f)$ cutting out $L(n,2n)$ in the Plücker embedding of $G(n,2n)$.
• To analyze the structure of the defining linear system separately for even and odd $n$, with the odd case deduced from the even case.

Proposed method

• Use a symplectic basis $\{e_1, \dots, e_{2n}\}$ with $\langle e_i, e_{2n-i+1} \rangle = 1$ and zero otherwise.
• Express elements of $\bigwedge^n E$ in Plücker coordinates $p_\alpha$ indexed by $n$-tuples $\alpha \in I(n,2n)$.
• Define the contraction map $f$ via $f(w) = \sum_{i<j} \langle w_i, w_j \rangle \widehat{w}_i \widehat{w}_j$, and characterize $\ker f$ via linear equations in Plücker coordinates.
• Introduce the linear forms $\Pi_{\alpha_{st}} = \sum_i c_{i,\alpha_{st},2n-i+1} X_{i,\alpha_{st},2n-i+1}$, where coefficients depend on support conditions.
• Use the decomposition $I(n-2,2n) = \bigcup_{i=1}^n (i) \times C_4(\Sigma(i)) \cup \text{other index sets}$ to organize the defining equations.
• Construct the matrix $B = \varphi(I(5,14))$ for $n=7$ as a direct sum of submatrices $\mathcal{L}_4, \mathcal{L}_3, \mathcal{L}_2$, corresponding to different index types.

Experimental results

Research questions

• RQ1 What is the explicit system of linear equations in Plücker coordinates that cut out the Lagrangian-Grassmannian $L(n,2n)$ over an arbitrary field?
• RQ2 How do the contraction maps induced by the symplectic form translate into linear conditions on Plücker coordinates?
• RQ3 Can the defining linear system for $\ker f$ be decomposed and classified by combinatorial types of index sets?
• RQ4 How does the structure of the defining equations differ between even and odd $n$?
• RQ5 What is the rank of the linear system $\ker f$ over fields of different characteristics?

Key findings

• The Lagrangian-Grassmannian $L(n,2n)$ is cut out in the Plücker embedding of $G(n,2n)$ by the quadratic Plücker relations and the linear forms $\Pi_{\alpha_{st}}$, which vanish precisely when $w \in \ker f$.
• For $n=7$, the defining linear system has $\text{rank}(B) = 2002$ over $\text{char}(F) = 0$ or $p \geq 5$, and $\text{rank}(B) = 1666$ in characteristic 2.
• The matrix $B$ decomposes into 7 copies of $\mathcal{L}_4$, 301 copies of $\mathcal{L}_3$, and 693 copies of $\mathcal{L}_2$, corresponding to different index configurations.
• The coefficients $c_{i,\alpha_{st},2n-i+1}$ are 1 only when the set $\{i, \alpha_{st}, 2n-i+1\}$ has size $n$, ensuring support compatibility.
• The construction is valid over any field, including positive characteristic, and the linear equations are explicitly given in terms of Plücker coordinates.
• The result establishes a complete, explicit, and characteristic-independent description of $\mathbb{P}(\ker f)$ as a linear subspace of $\mathbb{P}^{N-1}$, with $N = \binom{2n}{n}$.