Generic representations of the finite general linear groups and the Steenrod algebra. III
type: Item
instance of: scholarly article
title: Generic representations of the finite general linear groups and the Steenrod algebra. III
zbMATH Open document ID: 0831.20057
DOI: 10.1007/BF00961668
author: Nicholas J. Kuhn
published in: \(K\)-Theory
publication date: 1996-02-13T00:00:00Z
review text: [For part II cf. ibid. 8, No. 4, 395-428 (1994; Zbl 0830.20065).] Let \({\mathcal F} (q)\) be the abelian category with objects the functors \(F:\) finite \({\mathbf F}_q\)-vector spaces \(\to{\mathbf F}_q\)-vector spaces and with morphisms the natural transformations. The groups \(\text{Ext}^s_{{\mathcal F}(q)}(F,G)\) are conjecturally \(\text{Ext}^s_{\text{GL}(V)}(F(V),G(V))\) for finite \(F\) and \(G\) and large \(V\). The author develops some tools for studying such groups. In particular, with \(S^n\) defined by \(S^n(V)=V^{\otimes n}/\Sigma_n\), and with \(\Phi:S^n\to S^{qn}\) the \(q\)th power map, the following ``vanishing theorem'' is proved: for all finite \(F\in{\mathcal F}(q)\) and \(s>0\), one has \[ \text{colim}\{\text{Ext}^s_{{\mathcal F}(q)}(F,S^n)@>\Phi>>\text{Ext}^s_{{\mathcal F}(q)}(F,S^{qn})@>\Phi>>\dots\}=0. \] As an application, the space of natural transformations from the \(m\)th symmetric invariant functor to the \(n\)th symmetric coinvariant functor is calculated for all \(m\) and \(n\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 790464
zbMATH Keywords: representations
zbMATH Keywords: Steenrod algebras
zbMATH Keywords: MacLane homology
zbMATH Keywords: topological Hochschild homology
zbMATH Keywords: Abelian categories
zbMATH Keywords: functors
zbMATH Keywords: natural transformations
zbMATH Keywords: symmetric invariant functors
zbMATH Keywords: symmetric coinvariant functors
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 790464