Abstract

In 2021, Mücksch and Röhrle introduced the notion of an accurate arrangement. Specifically, let $\mathcal{𝒜}$ be a free arrangement of rank $\ell$. They call $\mathcal{𝒜}$ accurate if for every $1\le d\le \ell$, the first $d$ exponents of $\mathcal{𝒜}$ — when listed in increasing order — are realized as the exponents of a free restriction of $\mathcal{𝒜}$ to some intersection of reflecting hyperplanes of $\mathcal{𝒜}$ of dimension $d$. In this paper, if in addition the flats involved can be chosen to form a flag, we call $\mathcal{𝒜}$ flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them being the one consisting of Coxeter arrangements which we study systematically. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of Mücksch and Röhrle shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.

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