Studies on quasisymmetric functions

Abstract

In 1983, Ira Gessel introduced the ring of quasisymmetric functions (QSym), an extension of the ring of symmetric functions and nowadays one of the standard examples of a combinatorial Hopf algebra. In this thesis, I elucidate three aspects of its theory: 1) Gessel's P-partition enumerators are quasisymmetric functions that generalize, and share many properties of, the Schur functions; their Hopf-algebraic antipode satisfies a simple and explicit formula. Malvenuto and Reutenauer have generalized this formula to quasisymmetric functions "associated to a set of equalities and inequalities". I reformulate their generalization in the handier terminology of double posets, and present a new proof and an even further generalization in which a group acts on the double poset. 2) There is a second bialgebra structure on QSym, with its own "internal" comultiplication. I show how this bialgebra can be constructed using the Aguiar-Bergeron- Sottile universal property of QSym by extending the base ring; the same approach also constructs the so-called "Bernstein homomorphism", which makes any connected graded commutative Hopf algebra into a comodule over this second bialgebra QSym. 3) I prove a recursive formula for the "dual immaculate quasisymmetric functions" (a certain special case of P-partition enumerators) conjectured by Mike Zabrocki. The proof introduces a dendriform algebra structure on QSym. Two further results appearing in this thesis, but not directly connected to QSym, are: 4) generalizations of Whitney's formula for the chromatic polynomial of a graph in terms of broken circuits. One of these generalizations involves weights assigned to the broken circuits. A formula for the chromatic symmetric function is also obtained. 5) a proof of a conjecture by Bergeron, Ceballos and Labbé on reduced-word graphs in Coxeter groups (joint work with Alexander Postnikov). Given an element of a Coxeter group, we can form a graph whose vertices are the reduced expressions of this element, and whose edges connect two reduced expressions which are "a single braid move apart". The simplest part of the conjecture says that this graph is bipartite; we show finer claims about its cycles.