On the socle of Weyl modules
type: Item
instance of: scholarly article
title: On the socle of Weyl modules
zbMATH Open document ID: 0713.20035
DOI: 10.1007/BF01190225
author: Johannes Grabmeier
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: For a field k of characteristic p (p prime or 0) and to each partition \(\lambda =(\lambda_ 1,...,\lambda_ n)\), \(\lambda_ 1\geq \lambda_ 2\geq...\geq \lambda_ n\geq 0\), of a natural number \(n\geq 1\) \textit{R. W. Carter} and \textit{G. Lusztig} constructed a module \(V_{\lambda,k}\), which they called Weyl module [Math. Z. 136, 193-242 (1974; Zbl 0298.20009)]. These modules have the important property that they have a unique simple factor module \(F_{\lambda,k}\). Fixing n, these modules give a full set of simple modules for the Schur algebra \(S_ k(n,n)\). In case k has infinitely many elements these results can be interpreted for the polynomial representation theory of the \(GL_ n(k)\). Regarded as modules for the Schur algebra no such restriction is necessary. \textit{S. E. Fettes} considered in [Ph.D. thesis, Univ. Mass. (1982)] the socle of Weyl modules for some particular partitions, all of which happen to be simple modules. We extend her ideas to derive a relation between the socle of \(V_{\lambda,k}\) and the socle of the dual \(S_ k^{\lambda \#}\) of the Specht modules \(S_ k^{\lambda}\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 4173248
zbMATH Keywords: partition
zbMATH Keywords: Weyl module
zbMATH Keywords: simple factor module
zbMATH Keywords: simple modules
zbMATH Keywords: Schur algebra
zbMATH Keywords: representation theory
zbMATH Keywords: socle of Weyl modules
zbMATH Keywords: Specht modules
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01190225
OpenAlex ID: W2032605865
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description: scientific article; zbMATH DE number 4173248