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

Abstract

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 S n , 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 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.