The University of Michigan Combinatorics Seminar
Fall 2002
September 13, 4:10-5:00, 3866 East Hall

Structure of the peak Hopf algebra

Samuel Hsiao



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 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 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 chain-enumeration functionals 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 Charney-Davis index, which appears to be an Eulerian analog of the flag f-vector.