Homomorphisms between Solomon's descent algebras
type: Item
instance of: scholarly article
title: Homomorphisms between Solomon's descent algebras
zbMATH Open document ID: 0770.20008
DOI: 10.1016/S0021-8693(05)80042-0
author: Adriano M. Garsia
author: Christophe Reutenauer
author: F. Bergeron
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: For a permutation \(\sigma = \sigma_ 1 \dots \sigma_ n\) let the descent set of \(\sigma\) be \(\text{Des}(\sigma) = \{i: 1 \leq i \leq n-1\) and \(\sigma_ i \geq \sigma _{i+1}\}\) and define the element \(D_{\subseteq S} = \sum_{\text{Des}(\sigma)}\sigma\), where \(S \subseteq \{1,\dots,n-1\}\). The linear span of these elements forms a \(2^{n-1}\)-dimensional subalgebra \(\Sigma_ n\) of the group algebra of the symmetric group \(\mathbb{Q}[S_ n]\), called Solomon's descent algebra. For any \(s\), \(0 < s < n\), the authors construct a surjective homomorphism \(\Delta_ s: \Sigma_ n \to \Sigma_{n-s}\) and an embedding \(\Gamma_ s: \Sigma_{n-s} \to \Sigma_ n\). The homomorphisms \(\Delta_ s\) are also defined as derivations of the free associative algebra \(\mathbb{Q}\langle t_ 1,t_ 2,\dots\rangle\) sending \(t_ i\) to \(t_{i-s}\) in case one identifies the basis element \(D_{\subseteq S}\) of \(\Sigma_ n\) with some word on the alphabet \(T = \{t_ 1,t_ 2,\dots\}\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 130644
zbMATH Keywords: permutation
zbMATH Keywords: descent set
zbMATH Keywords: group algebra
zbMATH Keywords: symmetric group
zbMATH Keywords: Solomon's descent algebra
zbMATH Keywords: surjective homomorphism
zbMATH Keywords: embedding
zbMATH Keywords: derivations
zbMATH Keywords: free associative algebra
reviewed by: Melih Boral
MaRDI profile type: Publication
cites work: Shuffles of permutations and the Kronecker product
cites work: A decomposition of Solomon's descent algebra
cites work: Q3952291
cites work: Q3659988
cites work: Q3947818
cites work: Lie elements and an algebra associated with shuffles
cites work: A Mackey formula in the group of a Coxeter group. With an appendix by J. Tits: Two properties of Coxeter complexes
full work available at URL: s0021-8693(05)80042-0
OpenAlex ID: W1996291448
description: scientific article; zbMATH DE number 130644