# Analogs of Schur functions for rank two Weyl groups obtained from grid-like posets

## TL;DR

This paper introduces a uniform combinatorial approach to study lattice models for rank two Weyl group characters, deriving properties like symmetry, unimodality, and explicit generating functions without case-by-case analysis.

## Contribution

It provides a completely uniform and elementary combinatorial derivation of properties of lattice models for rank two Weyl group characters, replacing previous case-dependent methods.

## Key findings

- Lattices are rank symmetric and unimodal.
- Rank generating functions have quotient-of-products formulas.
- Results are derived using elementary combinatorial reasoning.

## Abstract

In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J. Wildberger, studied certain distributive lattice models for the `Weyl bialternants' (aka `Weyl characters') associated with the rank two root systems/Weyl groups. These distributive lattices were uniformly described as lattices of order ideals taken from certain grid-like posets, although the arguments connecting the lattices to Weyl bialternants were case-by-case depending on the type of the rank two root system. Using this connection with Weyl bialternants, these lattices were shown to be rank symmetric and rank unimodal, and their rank generating functions were shown to have beautiful quotient-of-products expressions. Here, these results are re-derived from scratch using completely uniform and elementary combinatorial reasoning in conjunction with some combinatorial methodology developed elsewhere by the second listed author.