Homomorphisms between Solomon's Descent Algebras

In a previous paper (see Garsia-Reutenauer [2]), we have studied algebraic properties of the descent algebras Σ n , and shown how these are related to the canonical decomposition of the free Lie algebra corresponding to a version of Poincaré-Birkhoff-Witt's theorem. In the present paper, we study homomorphisms between these algebras Σ n. The existence of these homomorphisms was suggested by properties of some directed graphs, we constructed in [6P], describing the structure of the descent algebras. More precisely, examination of the graphs suggested the existence of homomorphisms Σ n −→ Σ n−s and Σ n −→ Σ n+s. We were then able to construct, for any s, (0 < s < n), a surjective homomorphism ∆ s : Σ n −→ Σ n−s and an embedding Γ s : Σ n−s −→ Σ n , which reflects these observations. The homomorphisms ∆ s may also be defined as derivation of the free associative algebra Qt 1 , t 2 ,. .. which sends t i on t i−s , if one identifies the basis element D ⊆S of Σ n with some word (coding S) on the alphabet T = {t 1 , t 2 ,. . .}. We show that this mapping is indeed an homomorphism, using the combinatorial description of the multiplication table of Σ n given in [2]. 0. Introduction In the algebra of the symmetric group Q[S n ], let us define the element D ⊆S = Des(σ)⊆S σ where Des(σ) denotes the descent set of σ and S ⊆ {1,. .. , n − 1}. In [7], Solomon shows that the linear span Σ n of these 2 n−1 elements forms a subalgebra of Q[S n ]. In fact, Solomon shows that this is the case for any finite Coxeter group. In a previous paper [2], we have studied algebraic propertics of Σ n , and shown how these are related to the canonical decomposition of the free Lie algebra corresponding to a version of Poincaré-Birkhoff-Witt's theorem. In particular, we were able to compute a complete family of orthogonal primitive indempotents E λ of Σ n (indexed by partitions of n). We also computed the dimensions of the quasi-ideals E λ Σ n E µ. In the present paper, we study homomorphisms between these algebras Σ n. The existence of these homomorphisms was suggested by properties of the directed graphs (see [2]) describing the …