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 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.