Antichain simplices.

*(English)*Zbl 1433.52012Summary: 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 |

##### References:

[1] | David Anick, Construction d’espaces de lacets et d’anneaux locaux ‘a s´eries de Poincar´eBetti non rationnelles,C. R. Acad. Sci. Paris S´er. A-B290(1980), A729-A732. · Zbl 0466.13006 |

[2] | Luchezar L. Avramov, Infinite free resolutions. InSix Lectures on Commutative Algebra, Mod. Birkh¨auser Class., Birkh¨auser Verlag, 2010, pp. 1-118. |

[3] | Matthias Beck and Sinai Robins,Computing the Continuous Discretely, Undergraduate Texts in Mathematics, Springer, 2nd edition, 2015. · Zbl 1339.52002 |

[4] | Benjamin Braun and Brian Davis, Rationality of Poincar´e series for a family of lattice simplices. Preprint, available athttps://arxiv.org/abs/1711.04206. |

[5] | Benjamin Braun, Robert Davis, and Liam Solus, Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices,Adv. in Appl. Math.100 (2018), 122-142. · Zbl 1396.52020 |

[6] | Benjamin Braun and Fu Liu,h∗-Polynomials with roots on the unit circle. to appear in Experimental Mathematics. Available athttps://arxiv.org/abs/1807.00105. |

[7] | W. Bruns, B. Ichim, T. R¨omer, R. Sieg, and C. S¨oger, Normaliz: Algorithms for rational cones and affine monoids, Available athttps://www.normaliz.uni-osnabrueck.de. |

[8] | Winfried Bruns, Bogdan Ichim, and Christof S¨oger, The power of pyramid decomposition in Normaliz,J. Symbolic Comput.74(2016), 513-536. · Zbl 1332.68298 |

[9] | Brian Davis, Predicting the integer decomposition property via machine learning. To appear inProceedings of the 2018 Summer Workshop on Lattice Polytopes at Osaka University. Available athttps://arxiv.org/abs/1807.08399. |

[10] | Tor Holtedahl Gulliksen, Massey operations and the Poincar´e series of certain local rings, Preprint series: Pure mathematics, University of Oslo, 1970. Available athttps://www. duo.uio.no/bitstream/handle/10852/45414/20150821182559730.pdf. |

[11] | Takayuki Hibi, Akihiro Higashitani, and Nan Li, Hermite normal forms andδ-vectors, J. Combin. Theory Ser. A119(2012), 1158-1173. · Zbl 1242.05012 |

[12] | SageMath Inc., Cocalc collaborative computation online, 2018.https://cocalc.com/. |

[13] | Ezra Miller and Bernd Sturmfels,Combinatorial Commutative Algebra, Vol. 227 of Graduate Texts in Mathematics, Springer-Verlag, 2005. · Zbl 1090.13001 |

[14] | Sam Payne, Ehrhart series and lattice triangulations,Discrete Comput. Geom.40 (2008), 365-376. · Zbl 1159.52017 |

[15] | Irena Peeva,Graded Syzygies, Vol. 14 ofAlgebra and Applications, Springer-Verlag, 2011. · Zbl 1213.13002 |

[16] | Liam Solus, Localh∗-polynomials of some weighted projective spaces. To appear in Proceedings of the 2018 Summer Workshop on Lattice Polytopes at Osaka University. Available athttps://arxiv.org/abs/1807.08223. |

[17] | Liam Solus, Simplices for numeral systems,Trans. Amer. Math. Soc.371(2019), 2089- 2107. · Zbl 1406.52024 |

[18] | The Sage Developers,SageMath, the Sage Mathematics Software System (Version 8.4), 2018. Available athttp://www.sagemath.org. |

[19] | N. J. A. Sloane et al.,The On-Line Encyclopedia of Integer Sequences, published electronically athttps://oeis.org, 2019. · Zbl 1044.11108 |

[20] | G. K. White, Lattice tetrahedra,Canad. J. Math.16(1964), 389-396. · Zbl 0124.02901 |

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