Non-commutative Pieri operators on posets

Abstract

We 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.

LaTeX 2e, 22 pages Minor corrections, updated references. Complete and final version, to appear in issue of J. Combin. Th. Ser. A dedicated to G.-C. Rota

References in corpus (1)

• Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions

Cited by in corpus (20)

• Quasisymmetric Schur functions
• The peak algebra and the descent algebras of types B and D
• Skew quasisymmetric Schur functions and noncommutative Schur functions
• A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions
• Algebraic Structures on Grothendieck Groups of a Tower of Algebras
• The peak algebra and the Hecke-Clifford algebras at $q=0$
• Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
• Coloured peak algebras and Hopf algebras
• A lift of Schur's Q-functions to the peak algebra
• Row-strict Quasisymmetric Schur Functions, Characterizations of Demazure Atoms, and Permuted Basement Nonsymmetric Macdonald Polynomials
• The Pieri rule for dual immaculate quasi-symmetric functions
• Peaks Sets of Classical Coxeter Groups
• Fomin-Greene monoids and Pieri operations
• Representation theory of 0-Hecke-Clifford algebras
• The $cd$-Index: A Survey
• The enriched $q$-monomial basis of the quasisymmetric functions
• NSym into Q_{\infty} is not a Hopf Map
• Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras
• (Weak) incidence bialgebras of monoidal categories
• Pieri's Formula for Generalized Schur Polynomials