Ideal Decomposition of Hyperplane Arrangements

Let $\mathcal{A}$ be an affine hyperplane arrangement, $L(\mathcal{A})$ its intersection poset, and $\chi_{\mathcal{A}}(t)$ its characteristic polynomial. This paper aims to propose combinatorial structures for the factorization of $\chi_{\mathcal{A}}(t)$. To this end, we introduce the notion of an ideal decomposition of $L(\mathcal{A})$ and use the M\"{o}bius algebra as a key tool to derive such a factorization. This concept provides a unified and substantial generalization of both the modular elements proposed by Stanley (1971) and the nice partitions proposed by Terao (1992). We also define modular ideals of $L(\mathcal{A})$, which yield a tensor decomposition of the Orlik-Solomon algebra of $\mathcal{A}$. We further show that every modular ideal can be realized as the intersection poset of some hyperplane arrangement.