Around Solomon's descent algebras

We study different problems related to the 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 $\lceil |S| / 2 \rceil$.