Cyclically Compatible Deformations of the Braid Arrangement

Abstract

We prove a characteristic-polynomial shift formula for two-sided extensions of cyclically compatible deformations of the braid arrangement. For a nonnegative integer matrix $M=(m_{ij})$ with zero diagonal, let ${\mathcal{A}}_M$ be the arrangement \[ x_i-x_j=s,\qquad 1\le i<j\le n,\quad s\in[-m_{ij},m_{ji}]_{\mathbb{Z}}. \] Given $\alpha ,\beta \in {\mathbb{N}}^n$, define its two-sided extension ${\mathcal{A}}_M(\alpha ,\beta )$ by replacing this interval with \[ [-m_{ij}-α_i-β_j,\, m_{ji}+α_j+β_i]_{\mathbb{Z}}. \] Call $M$ cyclically compatible if all pairwise distinct $a,b,c$ with $1\le a,b,c\le n$ satisfy \[ m_{ac}\le m_{ab}+m_{bc}+1. \] Under this condition, for the reduced characteristic polynomial $\tilde{\chi }(\mathcal{A},t)=\frac{\chi (\mathcal{A},t)}{t}$, we have \[ \widetildeχ(\mathcal{A}_M(α,β),t) = \widetildeχ(\mathcal{A}_M,t-|α|-|β|). \] The proof uses the finite-field method and a cyclic-gap enumeration formula. We also establish redistribution invariance, study weak-sum perturbations, and give applications to Shi, uniform interval, graphical, and Ferrers-type deformations.

We withdraw this version because the main result presented in Section 3 was previously established in the literature, and our manuscript did not properly acknowledge this prior work. We will revise the manuscript to correct the attribution and clarify its original contributions