Tropical decomposition of Young's partition lattice

TL;DR

This paper introduces a tropical geometric approach to decompose Young's partition lattice into symmetric chains, providing new insights and partial solutions to longstanding conjectures about its structure.

Contribution

It develops a novel tropical decomposition method for Young's lattice, yielding symmetric chain decompositions for generic partitions, advancing understanding of its combinatorial properties.

Findings

Decomposition of $L(m,n)$ into level sets using tropical polynomials

Elementary raising and lowering algorithms for subposets

Symmetric chain decomposition for generic partitions

Abstract

Young's partition lattice $L(m,n)$ consists of unordered partitions having $m$ parts where each part is at most $n$. Using methods from complex algebraic geometry, R. Stanley proved that $L(m,n)$ is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that $L(m,n)$ has a stronger property called symmetric chain decomposition. Despite many efforts, this conjecture has only been proved for $\min(m,n)\leq 4$. In this paper, we decompose $L(m,n)$ into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we find that the resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of $L(m,n)$ consisting of "sufficiently generic" partitions.