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

TL;DR

This paper introduces poset models for Weyl group analogs of symmetric and Schur functions, providing a combinatorial framework that generalizes classical symmetric functions and explores their structural properties.

Contribution

It develops the theory of splitting posets for Weyl bialternants, offering new combinatorial models and results independent of Lie algebra representation theory.

Findings

Splitting posets are rank symmetric and unimodal.

They have quotient-of-product expressions for rank generating functions.

Many splitting posets exist outside Lie theoretic contexts.

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 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.