J. Austral. Math. Soc.  72 (2002), 317-333
The module structure of Solomon's descent algebra

Dieter Blessenohl
  Mathematisches Seminar der Universitat
  Ludewig-Meyn-Str. 4
  D--24098 Kiel
  Germany
  blessenohl@math.uni-kiel.de
and
Hartmut Laue
  Mathematisches Seminar der Universitat
  Ludewig-Meyn-Str. 4
  D--24098 Kiel
  Germany
  laue@math.uni-kiel.de


Abstract
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 $\Delta_n$, $n\in \mathbb{N}_0$. Each irreducible $\Delta_n$-module is shown to occur in at most one Loewy section of any principal indecomposable $\Delta_n$-module. A precise condition for this occurrence and formulae for the Cartan numbers are obtained.
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