A decomposition of Grassmannian associated with a hyperplane arrangement

TL;DR

This paper extends a recent hyperplane arrangement-based decomposition of the Grassmannian to general arrangements, unifying various decompositions and classifying related subspace restrictions.

Contribution

It generalizes the hyperplane arrangement decomposition of the Grassmannian to all arrangements and establishes the consistency of multiple related decompositions.

Findings

Unified $ ext{A}$-matroid, $ ext{A}$-adjoint, and Schubert decompositions.

Classified $k$-restrictions of arrangements via two methods.

Extended decomposition results beyond essential arrangements.

Abstract

The Grassmannian, which is the manifold of all $k$-dimensional subspaces in the Euclidean space $\mathbb{R}^n$, was decomposed through three equivalent methods connecting combinatorial geometries, Schubert cells and convex polyhedra by Gelfand, Goresky, MacPherson and Serganova. Recently, Liang, Wang and Zhao discovered a novel decomposition of the Grassmannian via an essential hyperplane arrangement, which generalizes the first two methods. However, their work was confined to essential hyperplane arrangements. Motivated by their research, we extend their results to a general hyperplane arrangement $\mathcal{A}$, and demonstrate that the $\mathcal{A}$-matroid, the $\mathcal{A}$-adjoint and the refined $\mathcal{A}$-Schubert decompositions of the Grassmannian are consistent. As a byproduct, we provide a classification for $k$-restrictions of $\mathcal{A}$ related to all $k$-subspaces through two equivalent methods: the $\mathcal{A}$-matroid decomposition and the $\mathcal{A}$-adjoint decomposition.