Tropical decomposition of Young's partition lattice

Details Tropical decomposition of Young's partition lattice Tropical decomposition of Young's partition lattice

2011-11-08

arxiv.org/abs/1111.2064v1

Mathematics

Combinatorics

Scientific paper

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 use tropical polynomials derived from the secant varieties of the rational normal curve in projective space in order to construct a canonical decomposition of $L(m,n)$ into centered subposets which are symmetric chain orders if $m$ is sufficiently large. In particular, we obtain a symmetric chain decomposition for the subposet of $L(m,n)$ consisting of "sufficiently generic" elements.

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