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arXiv · 2407.20008

A visual approach to symmetric chain decompositions of finite Young lattices

Terrance Coggins·Robert W. Donley Jr.·Ammara Gondal·Arnav Krishna
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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.

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Terrance Coggins, Robert W. Donley Jr., Ammara Gondal, Arnav Krishna. 2024-07-29. A visual approach to symmetric chain decompositions of finite Young lattices. https://arxiv.org/abs/2407.20008

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