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Stable hyperplane arrangements

Toshio Oshima

TL;DR

The work addresses the classification of complex hyperplane arrangements with a stability condition under coordinate forgetting, formalized as $L(\\mathcal{A})=\\pi_i^{-1}(\\pi_i(L(\\mathcal{A})))$ for all $i$. It develops a decomposition strategy and reduces the problem to a base case in two variables, yielding an explicit parametric form in terms of roots of unity and auxiliary scalars, thereby characterizing stable arrangements as pullbacks of mirror hyperplanes of complex reflection groups of type $A$ or $B$. Additional results determine the $v$-closed vectors for these stable configurations, linking the combinatorics of intersection posets to linear constraints. The findings connect hyperplane arrangement theory with middle convolution and Pfaffian systems, offering insights relevant to KZ-type systems and rigidity phenomena in associated differential equations.

Abstract

We classify complex hyperplane arrangements $\mathcal A$ whose intersection posets $L(\mathcal A)$ satisfy $L(\mathcal A)=π_i^{-1}\circπ_i\bigl(L(\mathcal A)\bigr)$ for $i=1,\dots,n$. Here $π_i$ 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 $π_i\bigl(L(\mathcal A)\bigr)=\{π_i(S)\mid S\in L(\mathcal A)\}$. We show that such arrangements $\mathcal A$ arise as pullbacks of the mirror hyperplanes of complex reflection groups of type $A$ or $B$.

Stable hyperplane arrangements

TL;DR

The work addresses the classification of complex hyperplane arrangements with a stability condition under coordinate forgetting, formalized as for all . It develops a decomposition strategy and reduces the problem to a base case in two variables, yielding an explicit parametric form in terms of roots of unity and auxiliary scalars, thereby characterizing stable arrangements as pullbacks of mirror hyperplanes of complex reflection groups of type or . Additional results determine the -closed vectors for these stable configurations, linking the combinatorics of intersection posets to linear constraints. The findings connect hyperplane arrangement theory with middle convolution and Pfaffian systems, offering insights relevant to KZ-type systems and rigidity phenomena in associated differential equations.

Abstract

We classify complex hyperplane arrangements whose intersection posets satisfy for . Here denotes the projection from onto defined by that forgets the coordinate of , and . We show that such arrangements arise as pullbacks of the mirror hyperplanes of complex reflection groups of type or .

Paper Structure

This paper contains 4 sections, 11 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Classification of stable hyperplane arrangements
  3. Proof of Theorem \ref{['thm:stableAr']}
  4. A related result

Key Result

Lemma 2.1

(i) Let $S\in\mathcal{L}$ and $T,\,T'\in\mathcal{L}^{(\mathop{\mathrm{codim}}\nolimits S+1)}_{\subset S}$ with $T\ne T'$. Then $\mathcal{A}_T\cap \mathcal{A}_{T'}=\mathcal{A}_S$. (ii) The arrangement $\mathop{\mathrm{mc}}\nolimits_v\mathcal{A}$ is $v$-closed. (iii) For $S\in\mathcal{L}$, the subs (v) Assume that $\mathcal{A}$ is $v$-closed and that $S\in\mathcal{L}$ is not $v$-closed. Let $H\i

Theorems & Definitions (33)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Example 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 23 more