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$k$-Adjoint of Hyperplane Arrangements

Weikang Liang, Suijie Wang, Chengdong Zhao

Abstract

In this paper, we introduce the $k$-adjoint of a given hyperplane arrangement $\mathcal{A}$ associated with rank-$k$ elements in the intersection lattice $L(\mathcal{A})$, which generalizes the classical adjoint proposed by Bixby and Coullard. The $k$-adjoint of $\mathcal{A}$ induces a decomposition of the Grassmannian, which we call the $\mathcal{A}$-adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of $\mathcal{A}$. Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the $k$-dimensional restrictions of $\mathcal{A}$. Consequently, we establish the anti-monotonicity property of some combinatorial invariants, such as Whitney numbers of the first kind and the independece numbers.

$k$-Adjoint of Hyperplane Arrangements

Abstract

In this paper, we introduce the -adjoint of a given hyperplane arrangement associated with rank- elements in the intersection lattice , which generalizes the classical adjoint proposed by Bixby and Coullard. The -adjoint of induces a decomposition of the Grassmannian, which we call the -adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of . Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the -dimensional restrictions of . Consequently, we establish the anti-monotonicity property of some combinatorial invariants, such as Whitney numbers of the first kind and the independece numbers.

Paper Structure

This paper contains 6 sections, 9 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. $k$-Adjoint of Hyperplane Arrangement
  3. $\mathcal{A}$-Decompositions of the Grassmannian
  4. Three $\mathcal{A}$-Decompositions of the Grassmannian
  5. Three $\mathcal{A}$-Decompositions Are the Same
  6. Combinatorial invariants

Key Result

Theorem 1.2

For a given hyperplane arrangement $\mathcal{A}$ in $\mathbb{R}^n$, the $\mathcal{A}$-matroid decomposition, the $\mathcal{A}$-adjoint decomposition and the refined $\mathcal{A}$-Schubert decomposition are exactly the same decomposition of the Grassmannian.

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 15 more