Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras

This article is devoted to the study of several algebras which are related to symmetric functions, and which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable $H_n(0)$-modules are discussed, and the homological properties of $H_n(0)$ are computed for small $n$. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.