Stable hyperplane arrangements

Abstract

We classify complex hyperplane arrangements $\mathcal{A}$ whose intersection posets $L(\mathcal{A})$ satisfy $L(\mathcal{A})=\ddot{I}{€}_i^{-1}\circ \ddot{I}{€}_i(L(\mathcal{A}))$ for $i=1,\dots ,n$. Here $\ddot{I}{€}_i$ denotes the projection from ${\mathbb{C}}^n$ onto ${\mathbb{C}}^{n-1}$ defined by that forgets the coordinate $x_i$ of $(x_1,\dots ,x_n)\in {\mathbb{C}}^n$, and $\ddot{I}{€}_i(L(\mathcal{A}))=\{\ddot{I}{€}_i(S)\mid S\in L(\mathcal{A})\}$. We show that such arrangements $\mathcal{A}$ arise as pullbacks of the mirror hyperplanes of complex reflection groups of type $A$ or $B$.

9 pages