Reflection groups acting on their hyperplanes. (Q1048218)

Let \(V\) be a finite-dimensional \(\mathbb{C}\)-vector space and let \(W\subset\text{GL}(V)\) be a finite pseudo-reflection group with corresponding hyperplane arrangement \(\mathcal A\). Assume \(\mathcal A\) is essential. For each \(H\in\mathcal A\) let \(\alpha_H\) denote some linear form with kernel \(H\). Let \(\mathbb{C}\mathcal A\) denote the complex vector space with basis \(\nu_H\), \(H\in\mathcal A\), and define a linear map \(\Phi\colon\mathbb{C}\mathcal A\to S^2V^*\) by \(\Phi(\nu_H)=\alpha^2_H\). The author proves, if \(\mathcal A\) is an essential reflection arrangement, then the mapping \(\Phi\) is surjective if and only if \(\mathcal A\) is irreducible. Further, using a family of representations of the braid group of \(W\), the author shows that the action of \(W\) on \(\mathcal A\) is canonically related to other natural representations of \(W\).