Lagrangian fibers of Gelfand--Cetlin systems of $\mathrm{SO}(n)$-type

TL;DR

This paper analyzes the structure of Gelfand--Cetlin systems for co-adjoint orbits of SO(n), describing their polytopes, face structures, and classifying all Lagrangian fibers using combinatorial methods.

Contribution

It provides a detailed combinatorial description of Gelfand--Cetlin polytopes and fibers for SO(n), including classification of all Lagrangian fibers, advancing understanding of symplectic geometry in this context.

Findings

Described face structure of Gelfand--Cetlin polytopes for SO(n)

Established iterated bundle structure of Gelfand--Cetlin fibers

Classified all Lagrangian fibers in this setting

Abstract

In this paper, we study the Gelfand--Cetlin systems and polytopes of the co-adjoint $\mathrm{SO}(n)$-orbits. We describe the face structure of Gelfand--Cetlin polytopes and iterated bundle structure of Gelfand--Cetlin fibers in terms of combinatorics on the ladder diagrams. Using this description, we classify all Lagrangian fibers.