Theta Positivity in Lagrangian Grassmannian

TL;DR

This paper investigates the structure of the theta nonnegative part of the Lagrangian Grassmannian, revealing its topological properties, relationships with other positive structures, and conditions for nonnegativity.

Contribution

It introduces an orbital decomposition of the theta nonnegative part and establishes its homeomorphism to a closed ball, connecting it with generalized Plücker coordinates.

Findings

The theta nonnegative part admits an orbital decomposition.

It is homeomorphic to a closed ball.

It aligns with nonnegativity conditions of generalized Plücker coordinates.

Abstract

We study the theta nonnegative part of Lagrangian Grassmannian. We show that it admits an orbital decomposition and is homeomorphic to a closed ball. We compare it with other positive structures. We show that it contains several totally nonnegative parts of Lagrangian Grassmannian subject to certain choices of pinnings and agrees with the nonnegativity of the generalized Pl\"ucker coordinates.