Applications of quasisymmetric schur functions

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Quasisymmetric functions arise in a number of areas of mathematics. For example, in discrete geometry they arise as weight enumerators for labelled partially ordered sets, and generate specific well known functions such as skew Schur functions, and Stanley symmetric functions. Meanwhile in the arena of category theory they are the terminal object in the category of combinatorial Hopf algebras equipped with a zeta function. With regard to combinatorial probability they arise in the study of random permutations with respect to a certain distribution. They also arise in representation theory as characters of a degenerate quantum group. Consequently a deep understanding of quasisymmetric functions is important as knowledge about them will impact each of the above areas, plus other areas in which they arise.