Antichain Simplices

Abstract

To each lattice simplex $\Delta$ we associate a poset encoding the additive structure of lattice points in the fundamental parallelepiped for $\Delta$. When this poset is an antichain, we say $\Delta$ is antichain. To each partition $\lambda$ of $n$, we associate a lattice simplex $\Delta_\lambda$ 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 $\lambda$ is relatively prime to $n-1$. We use these characterizations to experimentally study $\Delta_\lambda$ for all partitions of $n$ with $n\leq 73$. We also investigate the structure of these posets when $\lambda$ has only one or two distinct parts. Finally, we explain how this work relates to Poincar\'e series for the semigroup algebra associated to $\Delta$, and we prove that this series is rational when $\Delta$ is antichain.