A visual approach to symmetric chain decompositions of finite Young lattices

TL;DR

This paper explores the symmetric chain decompositions of finite Young lattices, linking their order structure to geometric and root system representations, and provides visual descriptions for specific cases.

Contribution

It introduces a geometric perspective on Young lattices and fully describes Lindström's symmetric chain decompositions for the case when m=3.

Findings

Order structure of Young lattices corresponds to dilated n-simplex points

Provides visual descriptions of symmetric chain decompositions for L(3, n)

Establishes connections to root system weight diagrams

Abstract

The finite Young lattice $L(m, n)$ is rank-symmetric, rank-unimodal, and has the strong Sperner property. R. Stanley further conjectured that $L(m, n)$ admits a symmetric chain order. We show that the order structure on $L(m, n)$ is equivalent to a natural ordering on the lattice points of a dilated $n$-simplex, which in turn corresponds to a weight diagram for the root system of type $A_n$. Lindstr{\" o}m's symmetric chain decompositions for $L(3, n)$ are described completely through pictures.