Hecke Algebras, Difference Operators, and Quasi-Symmetric Functions

Abstract We define a new action of the symmetric group and its Hecke algebra on polynomial rings whose invariants are exactly the quasi-symmetric polynomials. We interpret this construction in terms of a Demazure character formula for the irreducible polynomial modules of a degenerate quantum group. We use the action of the generic Hecke algebras to define quasi-symmetric and noncommutative analogues of Hall–Littlewood functions. We show that these generalized functions share many combinatorial properties with the classical ones.  Nous introduisons de nouvelles actions du groupe symetrique et de son algebre de Hecke sur les polynomes, pour lesquelles les invariants sont les polynomes quasi-symetriques. Nous interpretons cette construction en termes de caracteres de Demazure d'un groupe quantique degenere. Nous utilisons l'action de l'algebre de Hecke generique pour definir des analogues quasi-symetriques et non commutatifs des fonctions de Hall–Littlewood. Nous montrons que ces fonctions generalisees ont un certain nombre de proprietes communes avec les fonctions classiques.

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