Decomposition of Hyperplane Arrangements: Algebra, Combinatorics, and its Geometric Realization

TL;DR

This work develops a unified framework connecting combinatorics, algebra, and geometry to factorize characteristic polynomials of affine hyperplane arrangements. By introducing ideal decompositions in finite ranked meet-semilattices, it shows that $χ(L,t)$ factors as $t^{c}\prod χ(I_i,t)$, with $c = r(L) - \sum r(I_i)$, and demonstrates how hyperplane arrangements form natural instances where nice partitions arise as ideal decompositions. The paper then extends modular-element factorization to affine settings via modular ideals, proving geometric realizations for these ideals and establishing connections to modular subarrangements and cones. In the algebraic realm, it generalizes Terao's Orlik-Solomon factorization by constructing a canonical isomorphism that splits $A(\mathcal{A})$ as a tensor product when a modular ideal is present, thereby linking combinatorial decompositions to both geometric realizability and OS-algebra structure. Altogether, the results provide a versatile toolkit for factorization phenomena across combinatorics, geometry, and algebra of hyperplane arrangements.

Abstract

Let $\mathcal{A}$ be an affine hyperplane arrangement, $L(\mathcal{A})$ its intersection poset, and $χ_{\mathcal{A}}(t)$ its characteristic polynomial. This paper aims to find combinatorial conditions for the factorization of $χ_{\mathcal{A}}(t)$ and investigate corresponding algebraic interpretations. To this end, we introduce the concept of ideal decomposition in a finite ranked meet-semilattice. Notably, it extends two celebrated concepts: modular element proposed by Stanley in 1971 and nice partition proposed by Terao in 1992. The main results are as follows. An ideal decomposition of $L(\mathcal{A})$ leads to a factorization of its characteristic polynomial $χ_{\mathcal{A}}(t)$, which is an extension of Terao's factorization under a nice partition. A special type of two-part ideal decomposition, modular ideal, is studied, which extends Stanley's factorization to affine hyperplane arrangements. We also show that any modular ideal of $L(\mathcal{A})$ has a geometric realization. Moreover, we extend Terao's factorization of the Orlik-Solomon algebra for central arrangements, originally induced by modular elements, to arbitrary affine hyperplane arrangements via the modular ideals.

Figures (5)

• Figure 1: Meet-semilattice $L$ with the ideal decomposition $\{I_1, I_2\}$
• Figure 2: $L(\mathcal{A})$ in Example \ref{['eg1']} with the ideal decomposition $\{I_1, I_2, I_3\}.$
• Figure 3: $L(\mathcal{A})$ in Example \ref{['exthm4']} with the ideal decomposition $\{I_1, I_2\}.$
• Figure 4: Geometric realizations of $I_1$ (left) and $I_2$ (right) in Example \ref{['thm4P1']}.
• Figure 5: The intersection lattice $L(\mathcal{B}_4)$ of the braid arrangement $\mathcal{B}_4$.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Example 2.2
• Example 2.3
• Lemma 2.4
• proof
• proof : Proof of Theorem \ref{['thm2']}
• Lemma 2.5