Homomorphisms between Solomon's descent algebras

Abstract

In a previous paper (see A. Garsia and C. Reutenauer ( Adv. in Math. 77 , 1989, 189–262)), 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 the Poincaré-Birkhoff-Witt 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 that we constructed in the previous paper (reference above) 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 Δ s : Σ n →Σ n−s and an embedding Γ s :Σ n−s →Σ n , which reflects these observations. The homomorphisms Δ s may also be defined as derivations of the free associative algebra Q 〈 t 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 a homomorphism, using the combinatorial description of the multiplication table of Σ n given in the previous paper (reference above).