Generic representations of the finite general linear groups and the Steenrod algebra. II
type: Item
instance of: scholarly article
title: Generic representations of the finite general linear groups and the Steenrod algebra. II
zbMATH Open document ID: 0830.20065
DOI: 10.1007/BF00961409
author: Nicholas J. Kuhn
published in: \(K\)-Theory
publication date: 1994-12-04T00:00:00Z
review text: [For part I cf. Am. J. Math. 116, No. 2, 327-360 (1994; Zbl 0813.20049).] This paper relates the category of generic representations over the finite field and the modular representation theory of the groups \(\text{GL}_n(F_q)\). Let \({\mathcal G}L_n(q)\) be the category of left \(F_q[\text{GL}_n(F_q)]\) modules and let \({\mathcal M}_n(q)\) be the category of left \(F_q[M_n(F_q)]\) modules. Using the ``recollement'' setting the author shows that there are short exact sequences of Abelian categories \({\mathcal G}L_n(q)\to{\mathcal M}_n(q)\to{\mathcal M}_{n-1}(q)\). The second map is induced by multiplication by the matrix obtained from the \((n-1)\times(n-1)\) identity matrix by adding a row and column of zeroes. Using \({\mathcal F}(q)\) to denote the category whose objects are the functors from finite dimensional \(F_q\) vector spaces to all \(F_q\) vector spaces the ``recollement'' method is used to construct functors \(c^\infty_n:{\mathcal M}_n(q)\to{\mathcal F}(q)\) which preserve monos, epis, and direct sums and for which additional properties are proved. The author relates this work to the conjecture that `stable \(K\)-theory equals Topological Hochschild Homology'.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 19B14
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 682605
zbMATH Keywords: finite general linear groups
zbMATH Keywords: Steenrod algebras
zbMATH Keywords: stable \(K\)-theory
zbMATH Keywords: topological Hochschild homology
zbMATH Keywords: recollement
zbMATH Keywords: category of generic representations
zbMATH Keywords: modular representations
zbMATH Keywords: short exact sequences of Abelian categories
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00961409
OpenAlex ID: W2062302824
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description: scientific article; zbMATH DE number 682605