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Regular Unimodular Triangulations of Reflexive IDP 2-Supported Weighted Projective Space Simplices

Abstract

For each integer partition \(\mathbf {q}\) with d parts, we denote by \(\Delta _{(1,\mathbf {q})}\) the lattice simplex obtained as the convex hull in \(\mathbb {R}^d\) of the standard basis vectors along with the vector \(-\mathbf {q}\). For \(\mathbf {q}\) with two distinct parts such that \(\Delta _{(1,\mathbf {q})}\) is reflexive and has the integer decomposition property, we establish a characterization of the lattice points contained in \(\Delta _{(1,\mathbf {q})}\). We then construct a Gröbner basis with a squarefree initial ideal of the toric ideal defined by these simplices. This establishes the existence of a regular unimodular triangulation for reflexive 2-supported \(\Delta _{(1,\mathbf {q})}\) having the integer decomposition property. As a corollary, we obtain a new proof that these simplices have unimodal Ehrhart \(h^*\)-vectors.

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References

  1. 1.

    Gabriele Balletti, Takayuki Hibi, Marie Meyer, and Akiyoshi Tsuchiya. Laplacian simplices associated to digraphs. Arkiv för matematik, 56, 12 2018.

  2. 2.

    M. Beck and S. Robins. Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer New York, 2007.

  3. 3.

    Matthias Beck and Sinai Robins. Computing the continuous discretely. Undergraduate Texts in Mathematics. Springer, New York, second edition, 2015. Integer-point enumeration in polyhedra, With illustrations by David Austin.

  4. 4.

    Benjamin Braun. Unimodality problems in Ehrhart theory. In Recent trends in combinatorics, volume 159 of IMA Vol. Math. Appl., pages 687–711. Springer, [Cham], 2016.

  5. 5.

    Benjamin Braun and Robert Davis. Ehrhart series, unimodality, and integrally closed reflexive polytopes. Ann. Comb., 20(4):705–717, 2016.

  6. 6.

    Benjamin Braun, Robert Davis, and Liam Solus. Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices. Advances in Applied Mathematics, 100, 08 2018.

  7. 7.

    Benjamin Braun and Fu Liu. \(h^*\)-polynomials with roots on the unit circle. Experimental Mathematics, pages 1–17, 2019.

  8. 8.

    Winfried Bruns and Tim Römer. \(h\)-vectors of Gorenstein polytopes. J. Combin. Theory Ser. A, 114(1):65–76, 2007.

  9. 9.

    Heinke Conrads. Weighted projective spaces and reflexive simplices. Manuscripta Math., 107(2):215–227, 2002.

  10. 10.

    David A. Cox, John Little, and Donal O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics). Springer-Verlag, Berlin, Heidelberg, 2007.

  11. 11.

    Takayuki Hibi and Hidefumi Ohsugi. Quadratic initial ideals of root systems. Proc. Amer. Math. Soc., 130(7):1913–1922, 2001.

  12. 12.

    Fu Liu and Liam Solus. On the relationship between Ehrhart unimodality and Ehrhart positivity. Ann. Comb., 23(2):347–365, 2019.

  13. 13.

    Benjamin Nill. Volume and lattice points of reflexive simplices. Discrete Comput. Geom., 37(2):301–320, 2007.

  14. 14.

    Liam Solus. Simplices for numeral systems. Trans. Amer. Math. Soc., 371(3):2089–2107, 2019.

  15. 15.

    Richard P. Stanley. Combinatorics and commutative algebra, volume 41 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, second edition, 1996.

  16. 16.

    Bernd Sturmfels. Gröbner Bases and Convex Polytopes, volume 8 of University Lecture Series. American Mathematical Society, Providence, RI, 1996.

  17. 17.

    The Sage Developers. SageMath, the Sage Mathematics Software System (Version 8.9), 2019. https://www.sagemath.org.

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Correspondence to Benjamin Braun.

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BB was partially supported by NSF award DMS-1953785. DH was partially supported by NSF award DUE-1356253.

Communicated by Liam Solus

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Braun, B., Hanely, D. Regular Unimodular Triangulations of Reflexive IDP 2-Supported Weighted Projective Space Simplices. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00554-3

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Mathematics Subject Classification

  • Primary 52B20