The double Coxeter arrangement
type: Item
instance of: scholarly article
title: The double Coxeter arrangement
zbMATH Open document ID: 0949.52009
author: Louis Solomon
author: Hiroaki Terao
published in: Commentarii Mathematici Helvetici
publication date: 2000-11-20T00:00:00Z
review text: Consider a finite collection \(\mathcal A\) of linear hyperplanes in \({\mathbb R}^\ell\). Let \(\alpha_H: {\mathbb R}^\ell \to {\mathbb R}\) satisfy \(H=\ker \alpha_H\), for \(H\in {\mathcal A}\). Let \(S={\mathbb R}[x_1,\ldots,x_\ell]\). A derivation \(\theta\) of \(S\) is tangent along \(\mathcal A\) if \(\theta(\alpha_H)\) is a multiple of \(\alpha_H\) for every \(H\in {\mathcal A}\). The set \(D({\mathcal A})\) of all such derivations is an \(S\)-module. The arrangement \(\mathcal A\) is free if \(D({\mathcal A})\) is a free module, necessarily of rank \(\ell\). In this case a free basis for \(D({\mathcal A})\) may be chosen to be homogeneous, that is, the coefficients of \({{\partial}\over{\partial x_i}}\), \(1\leq i\leq \ell\) are homogeneous polynomials of the same degree. These degrees are called the exponents of the free arrangement \(\mathcal A\). Let \(W\subset GL(n,{\mathbb R})\) be a finite group generated by reflections. The collection \({\mathcal A}_W\) of mirror hyperplanes for all of the reflections in \(W\) is the Coxeter arrangement associated with \(W\). It is well-known that \({\mathcal A}_W\) is a free arrangement, with exponents coinciding with the exponents \(\deg(f_i)-1\), where \(\{f_1,\ldots, f_\ell\}\) is a set of basic invariant polynomials of \(W\). In this paper, the authors study the submodule \(E({\mathcal A}_W) \subset D({\mathcal A}_W)\) comprised of those derivations \(\theta\) for which \(\theta(\alpha_H)\) is a multiple of \(\alpha_H^2\) for each \(H\in {\mathcal A}\). It is shown that \(E({\mathcal A}_W)\) is a free \(S\)-module of rank \(\ell\), for any Coxeter arrangement \({\mathcal A}_W\). The coefficient polynomials occurring in a homogeneous free basis of \(E({\mathcal A}_W)\) all have degree equal to the Coxeter number \(h\) of \(W\). The motivation for the study of \(E({\mathcal A}_W)\) comes from a conjecture concerning Shi arrangements and an addition-deletion theorem involving arrangements with multiplicity (``multi…
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 05E15
zbMATH DE Number: 1180679
zbMATH Keywords: hyperplane arrangement
zbMATH Keywords: free arrangement
zbMATH Keywords: Shi arrangement
zbMATH Keywords: reflection group
zbMATH Keywords: basic invariant
zbMATH Keywords: Coxeter number
zbMATH Keywords: Jacobian matrix
reviewed by: Michael Falk
MaRDI profile type: Publication
full work available at URL: s000140050054
OpenAlex ID: W2085511025
DOI: 10.1007/S000140050054
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description: scientific article; zbMATH DE number 1180679