Florent Hivert

NONCOMMUTATIVE SYMMETRIC FUNCTIONS VI: FREE QUASI-SYMMETRIC FUNCTIONS AND RELATED ALGEBRAS

This article is devoted to the study of several algebras related to symmetric functions, 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 Hn(0)-modules are discussed, and the homological properties of Hn(0) are computed for small n. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.

The peak algebra and the Hecke-Clifford algebras at q = 0

Using the formalism of noncommutative symmetric functions, we derive the basic theory of the peak algebra of symmetric groups and of its graded Hopf dual. Our main result is to provide a representation theoretical interpretation of the peak algebra and its graded dual as Grothendieck rings of the tower of Hecke-Clifford algebras at q = 0.

Un analogue du monoı̈de plaxique pour les arbres binaires de recherche

Resume Nous introduisons une structure de monoide sur un ensemble darbres binaires etiquetes, par un procede analogue a la construction du monoide plaxique. Nous en deduisons une nouvelle approche de lalgebre des arbres binaires de Loday–Ronco. Pour citer cet article : F. Hivert et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 577–580.

Analogues non-commutatifs et quasi-symétriques des fonctions de Hall-Littlewood, et modules de Demazure d'une algèbre enveloppante quantique dégénérée

Resume Nous etablissons des formules des caracteres de type Weyl et Demazure pour certains groupes quantiques degeneres dont les caracteres sont des fonctions quasisymetriques. Nous en deduisons une action quasi-symetrisante de lalgebre de Hecke sur les polynomes, ce qui nous permet de definir des analogues quasi-symetriques et non-commutatifs des fonctions de Hall-Littlewood.

Une généralisation des fonctions quasi-symétriques et des fonctions symétriques non commutatives

Resume Nous construisons une bigebre auto-duale qui admet comme quotients ou comme sousalgebres, diverses generalisations recentes de lalgebre des fonctions symetriques, et dans laquelle on peut relever la plupart des constructions usuelles sur ces dernieres.

Some Generalizations of Quasi-symmetric Functions and Noncommutative Symmetric Functions

In this paper, we investigate various kinds of generalisations of symmetric functions. The classical algebra Sym of symmetric functions is embedded in QSym, the algebra of quasi-symmetric functions, and is also a quotient of the algebra Sym of noncommutative symmetric functions. A q-analogue QSym q of the algebra QSym provides a kind of unification of both generalizations QSym and Sym of Sym. In this paper we introduce some further genarlizations whose natural bases are various combinatorial objects like standard tableaux, permutations or integer matrices.