Canonical Hopf algebras between QSym and the peak algebra

   Sam Hsiao, (University of Michigan)

The algebra of quasisymmetric functions may be defined as the linear span of flag f-vector generating functions F(P) as P ranges over all graded posets. One might ask if any interesting subalgebras of QSym arise by restricting P to special families of graded posets. For example, it is well known that Stembridge's peak algebra arises from the class of Eulerian posets. I'll discuss an infinite family of Hopf subalgebras that interpolate between QSym and peak algebra. These are defined by letting P range over Ehrenborg's k-Eulerian posets. Many assertions about QSym and the peak algebra, such as freeness and universality, can be extended to these subalgebras. Relevance to open questions about flag f-vectors will be discussed.

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