Chevalley group theory and the transfer in the homology of symmetric groups
type: Item
instance of: scholarly article
title: Chevalley group theory and the transfer in the homology of symmetric groups
zbMATH Open document ID: 0606.20044
DOI: 10.1016/0040-9383(85)90001-1
author: Nicholas J. Kuhn
published in: Topology
publication date: 1985-01-01T00:00:00Z
review text: If L is a subgroup of a finite group G, let \(W_ G(L)\) denote the group \(N_ G(L)/L\) where \(N_ G(L)\) is the normalizer of L in G. There is a right action of \(W_ G(L)\) on the group cohomology \(H^*(L)\). The inclusion \(L\hookrightarrow G\) induces a map \(i^*: H^*(G)\to H^*(L)^{W_ G(L)}\) to the \(W_ G(L)\)-invariants. If K is a subgroup of G containing L, one has transfer maps \(tr^*: H^*(K)\to H^*(G)\) and \(\tau^*: H^*(L)^{W_ K(L)}\to H^*(L)^{W_ G(L)}\) induced by the inclusions \(K\hookrightarrow G\) and \(W_ K(L)\hookrightarrow W_ G(L).\) Starting with parabolic subgroups of a Chevalley group, the present paper studies cases for which one has \(i^*tr^*=\tau^*i^*\). The transfers \(\tau^*\) are studied in the context of the Hecke algebra, the particular case of the finite Lie groups theory context leads to a good understanding of the structure of natural maps between invariants of parabolic subgroups, as \(\tau^*.\) This paper, motivated by the establishment of connections between various works in stable homotopy theory, gives applications to topology, e.g. on inverse systems of Thom spectra, splittings using the Steinberg idempotent, James-Hopf invariants in homology.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55Q25
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20F40
zbMATH DE Number: 3979369
zbMATH Keywords: right action
zbMATH Keywords: group cohomology
zbMATH Keywords: invariants
zbMATH Keywords: transfer maps
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: Chevalley group
zbMATH Keywords: Hecke algebra
zbMATH Keywords: invariants of parabolic subgroups
zbMATH Keywords: stable homotopy
zbMATH Keywords: Thom spectra
reviewed by: Georges Hoff
MaRDI profile type: Publication
full work available at URL: 0040-9383(85)90001-1
OpenAlex ID: W2007301822
description: scientific article; zbMATH DE number 3979369