Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions
Nantel Bergeron and Stefan Mykytiuk and Frank Sottile and Stephanie van Willigenburg
In his work on $P$-partitions, Stembridge defined the algebra of peak functions $\Pi$, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show that $\Pi$ is closed under coproduct, and therefore a Hopf algebra, and describe the kernel of the retraction. Billey and Haiman, in their work on Schubert polynomials, also defined a new class of quasi-symmetric functions — shifted quasi-symmetric functions — and we show that $\Pi$ is strictly contained in the linear span $\Xi$ of shifted quasi-symmetric functions. We show that $\Xi$ is a coalgebra, and compute the rank of the $n$th graded component.