The University of Michigan Combinatorics Seminar
We consider graded representations of the algebra NC of non-commutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation are quasi-symmetric functions associated to intervals in P. We show these give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. We can extend this construction to arbitrary graphs. This provides a unified construction of quasi-symmetric functions that have appeared in different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the 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. We also show that the shifted quasi-symmetric functions of Billey and Haiman related to Schubert polynomials of all types, form a Hopf algebra.