On the Steinberg module, representations of the symmetric groups, and the Steenrod algebra
type: Item
instance of: scholarly article
title: On the Steinberg module, representations of the symmetric groups, and the Steenrod algebra
zbMATH Open document ID: 0593.20006
DOI: 10.1016/0022-4049(86)90147-7
author: Stephen A. Mitchell
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(A\) denote the mod \(p\) Steenrod algebra. Let \(B\) denote one of the subalgebras A(n) of \(A\) or \(P(n)\) of \(A/A\beta A\). \(B\) admits a left \(A\)-module structure extending its \(B\)-module structure. This result has independently been proved by the author [Topology 24, 227--246 (1985; Zbl 0568.55021)] and J. Smith. In the paper under review it is shown that for \(B=P(n)\) (or \(p=2\) and \(B=A(n))\) these two \(A\)-module structures are the same. The proof uses the fact that the Steinberg module is isomorphic to the Weyl module associated to the partition \(\Delta_ n=((p-1)(n-1),(p-1)(n-2),\ldots,p-1)\). Combining this Weyl module characterization of the Steinberg module with an unpublished result of S. Piddy, the author can deduce the following theorem of J. Smith: Let \(f\) denote the classical primitive idempotent corresponding to \(\Delta_ n\). Then \((\otimes^{(p-1)\binom{n}{2}}\mathbb Z_ n)f\) is a free \(P(n-2)\)-module on one generator.
reviewed by: Michael Clausen
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3953040
zbMATH Keywords: mod p Steenrod algebra
zbMATH Keywords: Steinberg module
zbMATH Keywords: Weyl module
zbMATH Keywords: partition
zbMATH Keywords: primitive idempotent
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90147-7
OpenAlex ID: W2067588085
cites work: On the modular representations of the general linear and symmetric groups
cites work: The representation theory of the symmetric groups
cites work: Finite complexes with A(n)-free cohomology
cites work: On the associated graded ring of a group ring
cites work: Q5619224
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description: scientific article; zbMATH DE number 3953040