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Antichain simplices. (English) Zbl 1433.52012
Summary: Associated with each lattice simplex \(\Delta\) is 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. For each partition \(\lambda\) of \(n\), we define 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 \(n \leq 73\). Further, we experimentally study the prevalence of the antichain property among simplices with a restricted type of Hermite normal form, suggesting that the antichain property is common among simplices with this restriction. Finally, we explain how this work relates to Poincaré series for the semigroup algebra associated with \(\Delta\) and we prove that this series is rational when \(\Delta\) is antichain.
MSC:
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
13D02 Syzygies, resolutions, complexes and commutative rings
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
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