The Shi arrangements and the Bernoulli polynomials
type: Item
instance of: scholarly article
title: The Shi arrangements and the Bernoulli polynomials
zbMATH Open document ID: 1251.32022
author: D. Suyama
author: Hiroaki Terao
published in: Bulletin of the London Mathematical Society
publication date: 2012-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.3214
Mathematics Subject Classification ID: 32S22
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 6048040
zbMATH Keywords: Coxeter arrangements
zbMATH Keywords: Shi arrangements
zbMATH Keywords: Bernoulli polynomials
MaRDI profile type: Publication
OpenAlex ID: W3102760931
arXiv ID: 1103.3214
DOI: 10.1112/BLMS/BDR118
review text: The braid arrangement is the Coxeter arrangement of type \(A_\ell\). The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations, see [\textit{J.-Y. Shi}, Lect. Notes Math. 1179. Berlin etc.: Springer (1986; Zbl 0582.20030)]. By coning the Shi arrangement, the authors have a central arrangement, denoted \(S_\ell\); its defining polynomial is given by NEWLINENEWLINE\[NEWLINEQ(S_\ell) = z \prod_{1 \leq p <w\leq \ell +1} (x_p -x_q) \prod_{1 \leq p <q \leq \ell+1}(x_p -x_q -z) = 0.NEWLINE\]NEWLINENEWLINE Using the Bernoulli polynomials, the authors construct, for the first time, an explicit basis for the derivation module of the cone over the Shi arrangement. This basis is given in the paper by the following definition, where \(D(S_\ell)\) denotes the derivation module over \(S_\ell\). NEWLINENEWLINENEWLINE Definition 3.1. Let \(\partial_i\) (\(1 \leq i \leq \ell +1\)) and \(\partial_z\) denote \(\partial/\partial x_i\) and \(\partial/\partial z\), respectively. Define homogeneous derivations NEWLINENEWLINE\[NEWLINE\eta_1:=\sum_{i=1}^{\ell+1} \partial_i \in D(S_\ell), \;\;\eta_2:=z\partial_z + \sum_{i=1}^{\ell+1}x_i\partial_i \in D(S_\ell),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\phi_j:=(x_j-x_{j+1}-z)\sum_{i=1}^{\ell+1}\sum_{0 \leq k_1\leq j-1, 0 \leq k_2 \leq \ell-j} (-1)^{k_1 + k_2} \sigma_{j-1-k_1}^{(j,1)} \sigma_{\ell-j-k_2}^{(j,2)} \bar B_{k_1, k_2}(x_i, z) \partial_iNEWLINE\]NEWLINE NEWLINEfor \(1\leq j \leq \ell\). NEWLINENEWLINENEWLINE In the above, \(\bar B_{k_1, k_2}(x_i, z) \) is determined via Bernoulli polynomials. Moreover, \(\sigma_k^{(j,s)}\) denotes the elementary symmetric function in the variables in \(I_s^{(j)}\) of degree \(k\) (\(s = 1,2\) and \(k\) is a nonnegative integer), where \(I_1^{(j)}:=\{x_1, \dots, x_{j-1}\},\;I_2^{(j)}:=\{x_{j+2}, \dots, x_{\ell+1}\}\).NEWLINENEWLINEThe authors demonstrate that this is a basis for \(A_3\) and conclude their paper with so…
reviewed by: Kelly J. Pearson
description: scientific article; zbMATH DE number 6048040