A symmetry of the descent algebra of a finite Coxeter group.
type: Item
instance of: scholarly article
title: A symmetry of the descent algebra of a finite Coxeter group.
zbMATH Open document ID: 1085.20018
author: D. Blessenohl
author: Christophe Hohlweg
author: Manfred Schocker
published in: Advances in Mathematics
publication date: 2005-06-10T00:00:00Z
review text: Let \((W,S)\) be a finite Coxeter system. For each \(I\subset S\), let \(W_I\) be the parabolic subgroup of \(W\) generated by \(I\), then \(W^I=\{x\in W\mid l(x)<l(xs)\) for all \(s\in I\}\) is a cross-section of the left cosets of \(W_I\) in \(W\), consisting of the unique representatives of minimal length. The linear span \(\mathcal D_W\) of the elements \(X^I=\sum_{x\in W^I}x\) (\(I\subset S\)) is a subalgebra of the integral group algebra \(\mathbb{Z} W\), the descent algebra of \(W\). Furthermore, if \(\varphi_I\) denotes the character of \(W\) induced by the principal character of \(W_I\), then the linear map \(\theta\) defined by \(\theta(X_I)=\varphi_I\) for all \(I\subset S\) is a homomorphism of rings mapping the descent algebra into the ring of class functions of \(W\). For \(W\) of type \(A\), \textit{A.~Jöllenbeck} and \textit{C. Reutenauer} derived the identity \(\theta(X)(Y)=\theta(Y)(X)\) for all \(X,Y\in\mathcal D_W\) [in Abh. Math. Semin. Univ. Hamb. 71, 105-111 (2001; Zbl 0996.05118)], where class functions of \(W\) have been extended to the group algebra of \(W\) linearly. They conjectured that this symmetry property of \(\mathcal D_W\) holds for arbitrary finite Coxeter groups \(W\). In the paper under review the authors prove that this conjecture on this symmetry property of \(\mathcal D_W\) holds for arbitrary finite Coxeter groups \(W\). As a consequence, several properties of the characters of \(W\) afforded by the primitive idempotents of \(\mathcal D_W\) are derived by the authors at once, including a symmetry of the corresponding character table, and a combinatorial description of their intertwining numbers with the descent characters of \(W\).
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 51F15
zbMATH DE Number: 2174976
zbMATH Keywords: Coxeter groups
zbMATH Keywords: descent algebras
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: coset representatives
zbMATH Keywords: integral group rings
zbMATH Keywords: representation rings
zbMATH Keywords: characters
MaRDI profile type: Publication
full work available at URL: j.aim.2004.05.007
OpenAlex ID: W2073898235
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DOI: 10.1016/J.AIM.2004.05.007
description: scientific article; zbMATH DE number 2174976