Antichain Simplices

Abstract

To each lattice simplex $Δ$ we associate a poset encoding the additive structure of lattice points in the fundamental parallelepiped for $Δ$. When this poset is an antichain, we say $Δ$ is antichain. To each partition $λ$ of $n$, we associate a lattice simplex $Δ_λ$ having one unimodular facet, and we investigate their associated posets. We give a number-theoretic characterization of the relations in these posets, as well as a simplified characterization in the case where each part of $λ$ is relatively prime to $n-1$. We use these characterizations to experimentally study $Δ_λ$ for all partitions of $n$ with $n\leq 73$. We also investigate the structure of these posets when $λ$ has only one or two distinct parts. Finally, we explain how this work relates to Poincaré series for the semigroup algebra associated to $Δ$, and we prove that this series is rational when $Δ$ is antichain.