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Inductive Freeness of Ziegler's Canonical Multiderivations for Restrictions of Reflection Arrangements

Torsten Hoge, Gerhard Roehrle, Sven Wiesner

TL;DR

This work classifies when Ziegler restrictions of restrictions of irreducible reflection arrangements are inductively free. Building on Ziegler's original freeness and a 2024 result showing inductive freeness persists under Ziegler restriction, the authors give a complete classification for $\mathscr A(W)^X$: the Ziegler restriction $(\mathscr A'', \kappa)$ is inductively free exactly when one of three conditions holds, corresponding to the inductive freeness of $\mathscr A(W)$, the monomial case $W = G(r,r,\ell)$ with the restriction inductively free, or the exceptional types $G_{33}$ or $G_{34}$ with rank at most 4. The proof combines analysis of the monomial and exceptional families, employs Euler multiplicities, and uses computer-assisted verification in intricate rank-4 scenarios. The results extend the understanding of the interplay between reflection groups, their restrictions, and multiarrangement freeness, with implications for the combinatorial and geometric structure of these arrangements.

Abstract

Let $\mathcal A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $\mathcal A"$ of $\mathcal A$ to any hyperplane endowed with the natural multiplicity $κ$ is then a free multiarrangement. In 2024, the first two authors proved an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if $\mathcal A$ is inductively free, then so is the free multiarrangement $(\mathcal A'',κ)$. In 2018, all reflection arrangements which admit inductively free Ziegler restrictions were classified by the first two authors. The aim of this paper is an extension of this classification to all restrictions of reflection arrangements utilizing the aforementioned fundamental result from the 2024 paper of the first two authors.

Inductive Freeness of Ziegler's Canonical Multiderivations for Restrictions of Reflection Arrangements

TL;DR

This work classifies when Ziegler restrictions of restrictions of irreducible reflection arrangements are inductively free. Building on Ziegler's original freeness and a 2024 result showing inductive freeness persists under Ziegler restriction, the authors give a complete classification for : the Ziegler restriction is inductively free exactly when one of three conditions holds, corresponding to the inductive freeness of , the monomial case with the restriction inductively free, or the exceptional types or with rank at most 4. The proof combines analysis of the monomial and exceptional families, employs Euler multiplicities, and uses computer-assisted verification in intricate rank-4 scenarios. The results extend the understanding of the interplay between reflection groups, their restrictions, and multiarrangement freeness, with implications for the combinatorial and geometric structure of these arrangements.

Abstract

Let be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction of to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. In 2024, the first two authors proved an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if is inductively free, then so is the free multiarrangement . In 2018, all reflection arrangements which admit inductively free Ziegler restrictions were classified by the first two authors. The aim of this paper is an extension of this classification to all restrictions of reflection arrangements utilizing the aforementioned fundamental result from the 2024 paper of the first two authors.
Paper Structure (10 sections, 30 theorems, 23 equations, 7 tables)

This paper contains 10 sections, 30 theorems, 23 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Recollections and Preliminaries
  3. Freeness of hyperplane arrangements
  4. Freeness of multiarrangements
  5. Reflection arrangements and their restrictions
  6. Proof of Theorem \ref{['thm:kappa2']}
  7. The monomial groups $G(r,r,n)$
  8. Groups of type $G_{33}$ and $G_{34}$
  9. Proof for Theorem \ref{['thm:kappa2']}
  10. Proof of Theorem \ref{['thm:delta-indfree']}

Key Result

Theorem 1.2

Let ${\mathscr A}$ be a free arrangement with exponents $\exp {\mathscr A} = \{1, e_2, \ldots, e_\ell\}$. Let $H_0 \in {\mathscr A}$ and consider the restriction ${\mathscr A}"$ with respect to $H_0$. Then the multiarrangement $({\mathscr A}", \kappa)$ is free with exponents $\exp ({\mathscr A}", \k

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2: ziegler:multiarrangements
  • Theorem 1.3: hogeroehrle:ZieglerII
  • Theorem 1.4: hogeroehrle:ZieglerII
  • Theorem 1.5: hogeroehrle:Ziegler
  • Corollary 1.6: hogeroehrle:Ziegler
  • Theorem 1.7
  • Corollary 1.8
  • Definition 1.9
  • Theorem 1.10: hogeroehrle:Ziegler
  • ...and 38 more