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The `Weyl symmetric functions' studied here naturally generalize classicalsymmetric (polynomial) functions, and `Weyl bialternants,' sometimes alsocalled Weyl characters, analogize the Schur functions. For this generalization,the underlying symmetry group is a finite Weyl group. A `splitting poset' for aWeyl bialternant is an edge-colored ranked poset possessing a certainstructural property and a natural weighting of its elements so that theweighted sum of poset elements is the given Weyl bialternant. Connected suchposets are of combinatorial interest in part because they are rank symmetricand rank unimodal and have nice quotient-of-product expressions for their rankgenerating functions. Supporting graphs of weight bases for irreduciblesemisimple Lie algebra representations provide one large family of examples.However, many splitting posets can be obtained outside of this Lie theoreticcontext. This monograph provides a tutorial on Weyl bialternants / Weylsymmetric functions and splitting posets that is largely self-contained andindependent of Lie algebra representation theory. New results are alsoobtained.

Poset models for Weyl group analogs of symmetric functions and Schur functions