The combinatorics of the bar resolution in group cohomology
type: Item
instance of: scholarly article
title: The combinatorics of the bar resolution in group cohomology
zbMATH Open document ID: 1057.57003
author: Victor Reiner
author: Peter Webb
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: This paper is devoted to explore the interaction between Combinatorics, Representation Theory and Topology. The work starts by introducing a double complex structure on the bar resolution used in computing the cohomology of a finite group and describing the two spectral sequences which arise from filtering it by rows and columns. This double complex structure turns out to be very closely related to the topology of the \(CW\)-complex \(K_{n}\) whose face poset is isomorphic to the subword ordering on words without repetition from an alphabet of size \(n\). This complex is shellable, and from its shellability it follows that the rank of its top homology is the number of permutations in the symmetric group \(S_{n}\) without fixed points, called derangements. The set of derangements in \(S_{n}\) is denoted by \(D_{n}\). Let us denote by \(L_{S}\) the fundamental quasisymmetric function on the subset \( S\subset [ n-1]=\{ 1,2,\dots ,n-1\} \) and define \( K_{n}( x) =\sum_{\pi \in D_{n}}L_{\text{Des}( \pi ) }\) where Des\((\pi) \) means the descent set of the permutation \(\pi \). The first significant result of this paper says that \(K_{n}( x) = \text{ch}( \varepsilon _{n}\otimes \chi _{n}) \), the Frobenius characteristic map of the homology representation of \(S_{n}\). Now, given a group \(G\), let \(\Delta \) be the simplicial complex whose vertices are the elements of \(G\), and in which every finite subset of \(G\) is a simplex. We may construct the oriented and ordered chain complexes denoted by \(C^{\text{ori}}( \Delta ) \) and \(C^{\text{ord}}( \Delta ) \), respectively, and one shows that there is a chain homotopy equivalence between them, so both of them can be used to compute the group homology \(H_{\ast }( G,M) \) with coefficients in any module \(M\). The associated double complex is defined as follows: we write \(C_{n,0}= \mathbb{Z}V_{n}\) where \(V_{n}\) is the diagonal of the \(( n+1) \) -fold product of copies of \(G\), \(C_{0,n}\) has a \(\mathbb{Z}\)-b…
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 55T99
zbMATH DE Number: 2093461
zbMATH Keywords: group cohomology
zbMATH Keywords: double complex
zbMATH Keywords: spectral sequence
reviewed by: Juan Antonio Pérez
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.12.006
OpenAlex ID: W2160890794
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DOI: 10.1016/J.JPAA.2003.12.006
description: scientific article; zbMATH DE number 2093461