dedicated to the memory of gian-carlo rotaWe consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu related to Eulerian posets is dual to the peak (Hopf ) algebra of Stembridge related to enriched P-partitions and connect this to the combinatorics of the Schubert calculus for isotropic flag manifolds.
Academic Press
In his work on P -partitions, Stembridge defined the algebra of peak functions Π, which is both a subalgebra and a retraction of the algebra of quasisymmetric functions. We show that Π is closed under coproduct, and therefore a Hopf algebra, and describe the kernel of the retraction. Billey and Haiman, in their work on Schubert polynomials, also defined a new class of quasi-symmetric functions -shifted quasi-symmetric functions -and we show that Π is strictly contained in the linear span Ξ of shifted quasi-symmetric functions. We show that Ξ is a coalgebra, and compute the rank of the nth graded component.Résumé. Dans ses travaux sur les P-partitions, Stembridge définit l'algèbre Π des fonctions de pics. Cette algèbre peutêtre vue comme une sous-algèbre ou un quotient de l'algèbre des fonctions quasi-symétriques. Nous montrons ici que Π est fermée sous le coproduit, et est donc une algèbre de Hopf. Nous décrivons aussi le noyau du quotient ci-dessus. D'autre part, dans leurs travaux sur les polynômes de Schubert, Billey et Haiman ont défini une nouvelle classe de fonctions quasi-symétriques: les fonctions quasi-symétrique décalé. Nous montrons que Π est strictement contenue dans l'espace linéaire Ξ des fonctions quasi-symétrique gauchis. Puis nous montrons que Ξ est une coalgèbre et calculons les dimensions des composantes de degrée n.
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