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 Σ 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 Σ. For the symmetric group Sn, there is a 1–1 correspondence between Σ and the set of all set compositions (or ordered set partitions) of {1,…,n}. The product on Σ has a simple combinatorial description in terms of set compositions. We study in detail the representation theory of the Solomon–Tits algebra of Sn over an arbitrary field, and show how our results relate to the corresponding results on the Solomon algebra of Sn. 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.