Descents and one-dimensional characters for classical Weyl groups
type: Item
instance of: scholarly article
title: Descents and one-dimensional characters for classical Weyl groups
zbMATH Open document ID: 0830.20025
DOI: 10.1016/0012-365X(93)E0179-8
author: Victor Reiner
published in: Discrete Mathematics
publication date: 1996-01-21T00:00:00Z
review text: This paper examines all sums of the form \(\sum_{\pi\in W}\chi(\pi)t^{d({\pi})}\) where \(W\) is a classical Weyl group, \(\chi\) is a character of degree 1 of \(W\) and \(d(\pi)\) is the descent statistic. The case when \(W\) is of type \(A_n\) (symmetric groups) has already been considered and involves the Eulerian and signed Eulerian polynomials [see \textit{J. Désarménien} and \textit{D. Foata}, Discrete Math. 99, 49-58 (1992; Zbl 0769.05094) and \textit{M. Wachs}, ibid. 99, 59-62 (1992; Zbl 0769.05098)]. This paper extends these results to the other classical Weyl groups of types \(B_n\) and \(D_n\). In fact, the answers turn out to be simpler and generalise further in these cases. In type \(B_n\) the proof emulates that of M. Wachs. For types \(B_n\) and \(D_n\), the sum factors completely into a product of simple linear terms when \(\chi\) is a non-trivial character.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 05E15
zbMATH DE Number: 769310
zbMATH Keywords: descents
zbMATH Keywords: one-dimensional characters
zbMATH Keywords: symmetric groups
zbMATH Keywords: signed Eulerian polynomials
zbMATH Keywords: classical Weyl groups
MaRDI profile type: Publication
cites work: The signed Eulerian numbers
cites work: Algebraic groups and modular Lie algebras
cites work: Opérations sur l'homologie cyclique des algèbres commutatives. (Operations on the cyclic homology of commutative algebras)
cites work: Signed permutation statistics
cites work: Binomial posets, Möbius inversion, and permutation enumeration
cites work: An involution for signed Eulerian numbers
description: scientific article; zbMATH DE number 769310