Samuel Hsiao
Samuel Hsiao

Ph.D. (2003) Cornell University

First Position
Dissertation
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Abstract: We study the algebraic and enumerative combinatorial aspects of Eulerian posets and their quasisymmetric generating functions. These generating functions span the socalled peak algebra Pi, which originated with Stembridge's theory of enriched Ppartitions. Remarkably, many constructs in the peak algebra that are natural in the context of enriched Ppartitions are also important from the viewpoint of flag enumeration. For example, we show that the fundamental basis of peak functions arising from enriched Ppartitions of chains is precisely the basis that is needed to properly encode the cdindex, a common invariant in the study of convex polytopes and Eulerian posets. As another example, the descentstopeaks map, which relates the ordinary and enriched theories of Ppartitions, turns out to be important for computing the flagenumerative information of an oriented matroid based on that of the underlying matroid.
We introduce a family of Eulerian posets, called posets of signed order ideals. Each of these posets is defined by appropriately "signing" a distributive lattice. We establish that the flagenumerative relationship between a poset of signed order ideals and its underlying distributive lattice is completely analogous to that between an oriented matroid and its underlying matroid. Furthermore, we show that these posets are ELshellable, they have nonnegative cdindices, and their quasisymmetric generating functions enumerate the enriched Ppartitions of certain labeled posets.
We analyze the (external) Hopf algebra structure on Pi, showing it to be dual to the concatenation Hopf algebra on letters of odd degree. Consequently, Pi is freely generated as a commutative algebra. Upon closer inspection of the generators, we find that Pi is free as a module over the ring of Schur Qfunctions. Much of our analysis builds on basic properties of a monomiallike basis for the peak algebra, as well as a recursively defined basis of eigenvectors for the descentstopeaks map. We give a simple criterion, in terms of this monomial basis, for determining when an element of Pi is symmetric.