Combinatorial Hopf algebras in the theory of symmetric functions

Symmetric functions form a commutative self-dual Hopf algebra based on the set of integer partitions. Its study leads naturally to the introduction of a wealth of Hopf algebras based on many kinds of combinatorial objects: compositions, permutations, tableaux, trees, parking functions, and many others. These algebras are often the same as those encountered in other fields, such as the theory of operads, or renormalization problems in quantum field theory. However, for applications to symetric functions, we need to realize these algebras in terms of an auxiliary set of variables. This leads to different constructions, relying upon analogues of the Robinson-Schensted correspondence and of the plactic monoid.