The peak algebra Pi was introduced by J. Stembridge in his development
of enriched Ppartitions. 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
posets.
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 Qfunction
algebra. These results mirror results on the structure of Qsym
and its
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
quasishuffles of peak compositions.
By duality, our results have implications for the algebra of
chainenumeration functionals
on Eulerian posets.
Earlier joint work with L. Billera and S. van Willigenburg identified the
cdindex 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
CharneyDavis index,
which appears to be an Eulerian analog of the flag fvector.
