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The module structure of Solomon's descent algebra

Part of: Representation theory of rings and algebras Algebraic combinatorics Representation theory of groups Radicals and radical properties of rings

Published online by Cambridge University Press:  09 April 2009

Dieter Blessenohl  and
Hartmut Laue
Dieter Blessenohl
Affiliation:
Mathematisches Seminar der Universität, Ludewig-Meyn-Str.4, D–24098 Kiel, Germany e-mail: blessenohl@math.uni-kiel.de, laue@math.uni-kiel.de
Hartmut Laue
Affiliation:
Mathematisches Seminar der Universität, Ludewig-Meyn-Str.4, D–24098 Kiel, Germany e-mail: blessenohl@math.uni-kiel.de, laue@math.uni-kiel.de
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Abstract

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A close connection is uncovered between the lower central series of the free associative algebra of countable rank and the descending Loewy series of the direct sum of all Solomon descent algebras Δn, n ∈ ℕ0. Each irreducible Δn-module is shown to occur in at most one Loewy section of any principal indecomposable Δn-module.A precise condition for his occurence and formulae for the Cartan numbers are obtained.

Keywords

descent algebraprincipal indecomposable modulesLoewy serieslower central seriesPoincar´-Birkhoff-Witt basesCartan numbersfree associative algebra

MSC classification

Secondary: 16N20: Jacobson radical, quasimultiplication 16G10: Representations of Artinian rings 20C30: Representations of finite symmetric groups 05E15: Combinatorial aspects of groups and algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 317 - 334
Copyright
Copyright © Australian Mathematical Society 2002

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