The module structure of the Solomon-Tits algebra of the symmetric group

Let $(W,S)$ be a finite Coxeter system. Tits defined an associative product on the set $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$. For the symmetric group $S_n$, there is a 1-1 correspondence between $\mathrm{\Sigma }$ and the set of all set compositions (or ordered set partitions) of $\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$. The product on $\mathrm{\Sigma }$ 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.