on lagrangian-grassmannian variety

TL;DR

This paper characterizes the linear relations that vanish the Lagrangian-Grassmannian and describes its Plücker matrix as a direct sum of specific matrices, advancing understanding of its algebraic structure.

Contribution

It proves that only linear combinations of the Family of Linear Relations of Contraction vanish the Lagrangian-Grassmannian and details the structure of its Plücker matrix.

Findings

Family of Linear Relations of Contraction uniquely vanish the Lagrangian-Grassmannian.

Plücker matrix decomposes into a direct sum of incidence, regular, and sparse matrices.

Entries of the Plücker matrix are only 0 or 1.

Abstract

In this paper it is shown that Family of Linear Relations of Contraction ($FRLC$) are the only ones, up to linear combination, that vanish the Lagrangian-Grassmannian. It is shown that the Pl\"ucker matrix of the Lagrangian-Grassmannian is a direct sum of incidence matrix, regular and sparce with entries in the set { 0, 1}.