# Lagrangian configurations and symplectic cross-ratios

## TL;DR

This paper explores the geometry of Lagrangian configurations in symplectic spaces, establishing their relation to difference operators and introducing symplectic cross-ratios with a key algebraic relation.

## Contribution

It introduces a novel connection between moduli spaces of Lagrangian configurations and symmetric difference operators, and defines symplectic cross-ratios satisfying a Pfaffian relation.

## Key findings

- Moduli spaces are isomorphic to quotients of symmetric difference operators.
- Symplectic cross-ratios parametrize configurations for N=2n+2.
- A Pfaffian relation links the cross-ratios and determinants.

## Abstract

We consider moduli spaces of cyclic configurations of $N$ lines in a $2n$-dimensional symplectic vector space, such that every set of $n$ consecutive lines generates a Lagrangian subspace. We study geometric and combinatorial problems related to these moduli spaces, and prove that they are isomorphic to quotients of spaces of symmetric linear difference operators with monodromy $-1$. The symplectic cross-ratio is an invariant of two pairs of $1$-dimensional subspaces of a symplectic vector space. For $N = 2n+2$, the moduli space of Lagrangian configurations is parametrized by $n+1$ symplectic cross-ratios. These cross-ratios satisfy a single remarkable relation, related to tridiagonal determinants and continuants, given by the Pfaffian of a Gram matrix.