Equivariant multiplicities of Coxeter arrangements and invariant bases
type: Item
instance of: scholarly article
title: Equivariant multiplicities of Coxeter arrangements and invariant bases
zbMATH Open document ID: 1267.52019
author: Takuro Abe
author: Hiroaki Terao
author: Atsushi Wakamiko
published in: Advances in Mathematics
publication date: 2012-07-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.0329
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 6060972
zbMATH Keywords: arrangement of hyperplanes
zbMATH Keywords: Coxeter arrangements
zbMATH Keywords: equivariant multiplicities
zbMATH Keywords: invariant bases
reviewed by: Kelly J. Pearson
MaRDI profile type: Publication
OpenAlex ID: W2095259160
arXiv ID: 1011.0329
cites work: A primitive derivation and logarithmic differential forms of Coxeter arrangements
cites work: Primitive filtrations of the modules of invariant logarithmic forms of Coxeter arrangements
cites work: The freeness of Shi-Catalan arrangements
cites work: Totally free arrangements of hyperplanes
cites work: Coxeter multiarrangements with quasi-constant multiplicities
cites work: Q4003206
cites work: Q3961117
cites work: On a linear structure of the quotient variety by a finite reflexion group
cites work: The double Coxeter arrangement
cites work: Multiderivations of Coxeter arrangements
cites work: Bases for the derivation modules of two-dimensional multi-Coxeter arrangements and universal derivations
cites work: The primitive derivation and freeness of multi-Coxeter arrangements.
cites work: Q3834612
DOI: 10.1016/J.AIM.2012.04.015
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review text: Let \({\mathcal A}\) be an irreducible Coxeter arrangement and \(W\) be its Coxeter group. For each \(H \in {\mathcal A}\), choose \(\alpha_H \in V^*\) with \(H =\text{ker}(\alpha_H) \in S\). A map \({\mathcal A} \to Z\) is called a multiplicity of \({\mathcal A}\). The multi-derivation module of \({\mathcal A}\) is the \(S\)-module \(D({\mathcal A}, {\mathbf m}) =\{\theta\mid \theta(\alpha_H) \in \alpha_H^{{\mathbf m}(H)} S_{(\alpha_H)} \text{\;for\;all\;} H \in {\mathcal A}\}\). A multiarrangement \(({\mathcal A}, {\mathbf m})\) is free if \(D({\mathcal A}, {\mathbf m}) = \bigoplus S \cdot \theta_i\) with homogeneous basis \(\theta_i\), where \(\deg (\theta_i) \) are called the exponents. A multiplicity \({\mathbf m}:{\mathcal A} \to Z\) is equivariant when \({\mathbf m}(H) = {\mathbf m}(\omega H)\) for any \(H\in {\mathcal A}\) and any \(\omega \in W\).NEWLINENEWLINEThis paper's main result generalizes the main theorem of \textit{H. Terao} [Invent. Math. 148, No. 3, 659--674 (2002; Zbl 1032.52013)].NEWLINENEWLINEMain Result: For any irreducible Coxeter arrangement \({\mathcal A}\) and any equivariant multiplicity, \textbf{m}, the multiarrangement \(({\mathcal A}, {\mathbf m})\) is free.NEWLINENEWLINEThe authors prove their result by giving an explicit construction of a basis for \(D({\mathcal A}, {\mathbf m})\) when \textbf{m} is equivariant.
description: scientific article; zbMATH DE number 6060972