The separator-based derivations for arrangements of monomial groups

Abstract

We investigate derivation modules of hyperplane arrangements associated with the monomial groups $G(r,p,\ell )$. Motivated by Mühlherr's separator-based construction for graphic arrangements, we extend this approach by combining separators with integral expressions. For a subarrangement ${\mathcal{A}}_G$ defined by a graph $G$, we show that, when $p<r$, the derivation module $\mathcal{D}({\mathcal{A}}_G)$ is generated by the standard derivations ${\eta }_0,\dots ,{\eta }_{\kappa (G)}$ together with a finite set of separator-based derivations. This result generalizes the complete graph case and provides new evidence toward the broader applicability of separator methods in the study of hyperplane arrangements.

arXiv comment17 pages