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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. |
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