The peak algebra Pi was introduced by J. Stembridge in his development
of enriched P-partitions. It is a Hopf subalgebra of Qsym,
the Hopf algebra of quasisymmetric functions. The Hopf structure has become
important in connecting Pi to the enumeration of chains in Eulerian
We will describe the structure of Pi, showing it to be
a free polynomial algebra, a cofree graded coalgebra,
and a free module over Schur's Q-function
algebra. These results mirror results on the structure of Qsym
relationship to the symmetric functions.
We introduce a new basis of monomial peak functions for Pi which
behaves much like the monomial basis for Qsym.
For example, the stucture constants relative to this new basis count
quasi-shuffles of peak compositions.
By duality, our results have implications for the algebra of
on Eulerian posets.
Earlier joint work with L. Billera and S. van Willigenburg identified the
cd-index as the dual basis
to Stembridge's basis of fundamental peak functions. Here we find that
the monomial basis for Pi is dual to N. Reading's
which appears to be an Eulerian analog of the flag f-vector.