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

Abstract

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 quotientof-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. In particular, a cancelling argument of Stembridge is reworked to provide sufficient combinatorial conditions for a given poset to be splitting. This new splitting theorem is used to help construct what are here named crystalline splitting posets. The Weyl bialternants with unique splitting posets are classified, and a new combinatorial characterization of splitting posets associated with the minuscule and quasi-minuscule dominant weights is given. These findings are supported by other new results of a poset-structural nature.In addition, some new conceptual approaches are presented. The notion of a "refined" splitting poset is introduced to address not only splitting-type problems but also Littlewood-Richardson-type (i.e. product decomposition) and branching-type problems. "Crystalline splitting posets" are introduced as a class of posets that behave much like crystal graphs from the crystal base theory of Kashiwara et al with regard to branching and decomposing products. A new technique called "vertex coloring" is used in conjunction with the new splitting theorem to produce crystalline splitting posets and, in particular, all crystal graphs. Via this vertexcoloring method, Stembridge's admissible systems are shown to be crystalline splitting posets.Crystal basis / crystal graph theory is used to demonstrate uniqueness of crystalline splitting posets obtained via a product construction.

Unified characterizations of minuscule Kac--Moody representations built from colored posets

“…Minuscule posets are the structures on which the Littlewood-Richardson and cohomology calculations for minuscule varieties performed in [BuSa] and its references are based. Donnelly has recently given [Do2] a new combinatorial characterization of the minuscule lattices of [Pr1] while developing a new version of crystal graphs.…”