Embeddings of higher Lie modules.
type: Item
instance of: scholarly article
title: Embeddings of higher Lie modules.
zbMATH Open document ID: 1048.20030
DOI: 10.1016/S0022-4049(03)00100-2
author: Manfred Schocker
published in: Journal of Pure and Applied Algebra
publication date: 2003-11-19T00:00:00Z
review text: Consider the general linear group \(\text{GL}(V)\) of a finite dimensional \(K\)-vector space \(V\) where \(K\) is a field of characteristic zero. The tensor algebra \(T(V)\) has the structure of both a Lie algebra and a \(\text{GL}(V)\)-module. For each positive integer \(n\), the homogeneous component \(T_n(V)\) is a \(\text{GL}(V)\)-module. Of interest here are certain \(\text{GL}(V)\)-submodules of \(T_n(V)\): the classical Lie module \(L_n(V)\) and higher Lie modules corresponding to partitions of \(n\). The Lie module \(L_n(V)=T_n(V)\cap L(V)\) where \(L(V)\) is the Lie subalgebra generated by \(V\). The higher Lie modules are defined in terms of the classical Lie modules and each \(T_n(V)\) decomposes as sum of higher Lie modules. Using idempotents in the group algebra of a symmetric group and a product formula for these idempotents found by \textit{D. Blessenohl} and \textit{H. Laue} [J. Algebra 180, No. 3, 698-724 (1996; Zbl 0864.20007)], the author shows that for \(n\geq 4\), there is a chain of embeddings of higher Lie modules for those corresponding to hook partitions. This chain of embeddings is then used to study the classical Lie modules. Each \(L_n(V)\) is the direct sum of simple modules of the form \(M_p\otimes_{KS_n}T_n(V)\) over partitions \(p\vdash n\). Here \(S_n\) is the symmetric group on \(n\) objects and \(M_p\) is the simple \(KS_n\)-module corresponding to the partition \(p\). Each \(M_p\) may appear multiple times, and the fundamental question is to determine precisely the multiplicity for each partition \(p\). For \(n\geq 3\), the author determines those partitions \(p\vdash n\) for which \(M_p\) has multiplicty at least one, thus recovering a result of \textit{A. A. Klyachko} [Sib. Math. J. 15, 914-920 (1975); translation from Sib. Mat. Zh. 15, 1296-1304 (1974; Zbl 0315.15015)]. Further, the author determines those partitions whose corresponding simple modules have multiplicity exactly one. The author ends the paper with a discussion o…
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 2005932
zbMATH Keywords: tensor algebras
zbMATH Keywords: Lie modules
zbMATH Keywords: decomposition multiplicities
zbMATH Keywords: general linear groups
zbMATH Keywords: idempotents
zbMATH Keywords: symmetric groups
zbMATH Keywords: hook partitions
Wikidata QID: Q115340134
MaRDI profile type: Publication
cites work: On the descending Loewy series of Solomon's descent algebra
cites work: The Free Lie Ring and Lie Representations of the Full Linear Group
cites work: Q5785190
cites work: Q3952291
cites work: Q4267501
cites work: A symmetry property of Solomon's algebra and of higher Lie characters
cites work: Lie elements in the tensor algebra
cites work: Q3134850
cites work: Die linearen Beziehungen zwischen höheren Kommutatoren
cites work: Q4236280
cites work: On Symmetrized Kronecker Powers and the Structure of the Free Lie Ring
cites work: Über Invarianten in Lieschen Ringen
cites work: The Eulerian representations of \(\Sigma_ n\) as restrictions of representations of \(\Sigma_{n+1}\)
cites work: Multiplicities of irreducible components of a free Lie algebra as a linear group module
description: scientific article; zbMATH DE number 2005932