Algebraic combinatorics related to the free Lie algebra, Séminaire Lotharingien de Combinatoire

During the past decade numerous fruitful contributions to the theory of the free Lie algebra have been made. Results and methods in this area are characterized by a subtle interplay between algebraic and combinatorial ideas. The quantity and the wealth of the material accumulated in the last years might be accompanied by the certainly undesired side-effect of concealing its own roots and history. We have therefore decided to restrict ourselves to a concise approach to some specially chosen topics. It is essentially self-contained and takes its course starting from a completely elementary source. At the same time the attempt is made to duly provide the reader with appropriate references for the various contributions involved. We take the opportunity to refer to [18] as a useful supplement to this article. In the first section we describe certain aspects of the theory and a number of known results. In the second section we show that these are different branches of the same key result (Ind) for which we then give a proof, by means of elementary combinatorial reasoning, in the third section. The underlying idea of our approach is to transfer problems on free Lie algebras into the area of group rings of symmetric groups. On the one hand, this provides a powerful tool to solve those problems. On the other hand, the arising questions are a challenging contribution t