  Let $(W,S)$ be a finite Coxeter system. Tits defined an associative product
on the set $\Sigma$ of simplices of the associated Coxeter complex. The
corresponding semigroup algebra is the Solomon-Tits algebra of $W$. It contains
the Solomon algebra of $W$ as the algebra of invariants with respect to the
natural action of $W$ on $\Sigma$. For the symmetric group $S_n$, there is a
1-1 correspondence between $\Sigma$ and the set of all set compositions (or
ordered set partitions) of $\{1,...,n\}$. The product on $\Sigma$ has a simple
combinatorial description in terms of set compositions. We study in detail the
representation theory of the Solomon-Tits algebra of $S_n$ over an arbitrary
field, and show how our results relate to the corresponding results on the
Solomon algebra of $S_n$. This includes the construction of irreducible and
principal indecomposable modules, a description of the Cartan invariants, of
the Ext-quiver, and of the descending Loewy series. Our approach builds on a
(twisted) Hopf algebra structure on the direct sum of all Solomon-Tits
algebras.
