  The `Weyl symmetric functions' studied here naturally generalize classical
symmetric (polynomial) functions, and `Weyl bialternants,' sometimes also
called Weyl characters, analogize the Schur functions. For this generalization,
the underlying symmetry group is a finite Weyl group. A `splitting poset' for a
Weyl bialternant is an edge-colored ranked poset possessing a certain
structural property and a natural weighting of its elements so that the
weighted sum of poset elements is the given Weyl bialternant. Connected such
posets are of combinatorial interest in part because they are rank symmetric
and rank unimodal and have nice quotient-of-product expressions for their rank
generating functions. Supporting graphs of weight bases for irreducible
semisimple Lie algebra representations provide one large family of examples.
However, many splitting posets can be obtained outside of this Lie theoretic
context. This monograph provides a tutorial on Weyl bialternants / Weyl
symmetric functions and splitting posets that is largely self-contained and
independent of Lie algebra representation theory. New results are also
obtained.
