Applied Algebra Seminar

York University - Fall 2002

September 23, 2002 - 4:30pm
Ross N638

Speaker: Samuel Hsiao

Cornell University



The peak Hopf algebra and some connections to enumeration in posets 



The peak algebra  $\Pi$ was introduced by J. Stembridge in his development
of enriched $P$-partitions.
It is a Hopf subalgebra of the quasisymmetric functions $\Qsym.$
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 $Q$-function
algebra. These results mirror results on the structure of $\Qsym$ and its
relationship to the symmetric functions.
We introduce a new basis of {\em 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 {\em peak compositions.}

By duality, our results have implications for the algebra of
chain-enumeration functionals on Eulerian posets.
Earlier joint work with L. Billera and S. van Willigenburg identified the
${\bf cd}$-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 Charney-Davis index,
which appears to be an Eulerian analog of the flag $f$-vector.

A knowledge of quasisymmetric functions and flag $f$-vectors is useful but
not required for this talk.


Algebra Seminar Home - Fall 2002