# Applied Algebra Seminar

## York University - Fall 2002

September 23, 2002 - 4:30pm
Ross N638

## Speaker: Samuel Hsiao

Cornell University

## Abstract:

 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.