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Around Solomon’s descent algebras. (English) Zbl 1193.20046
Summary: We study different problems related to Solomon’s descent algebra \(\Sigma(W)\) of a finite Coxeter group \((W,S)\): positive elements, morphisms between descent algebras, Loewy length…. One of the main result is that, if \(W\) is irreducible and if the longest element is central, then the Loewy length of \(\Sigma(W)\) is equal to \(\Bigl\lceil\tfrac{|S|}{2}\Bigr\rceil\).

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C30 Representations of finite symmetric groups
Software:
CHEVIE
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