Annals of Combinatorics

, Volume 22, Issue 4, pp 875–884

Volume, Facets and Dual Polytopes of Twinned Chain Polytopes

Article

Abstract

Let $$(P,\le _P)$$ and $$(Q,\le _Q)$$ be finite partially ordered sets with $$|P|=|Q|=d$$, and $$\mathcal {C}(P) \subset \mathbb {R}^d$$ and $$\mathcal {C}(Q) \subset \mathbb {R}^d$$ their chain polytopes. The twinned chain polytope of P and Q is the lattice polytope $$\Gamma (\mathcal {C}(P),\mathcal {C}(Q)) \subset \mathbb {R}^d$$ which is the convex hull of $$\mathcal {C}(P) \cup (-\mathcal {C}(Q))$$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes.

Keywords

Gorenstein Fano polytope Reflexive polytope Order polytope Chain polytope Volume Facet Dual polytope

52B05 52B20

References

1. 1.
V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom., 3 (1994), 493–535.
2. 2.
T. Chappell, T. Friedl and R. Sanyal, Two double poset polytopes, SIAM J. Discrete Math., 31 (2017), no. 4, 2378–2413.Google Scholar
3. 3.
D. Cox, J. Little and H. Schenck, “Toric Varieties”, Amer. Math. Soc., 2011.Google Scholar
4. 4.
T. Hibi, “Algebraic Combinatorics on Convex Polytopes”, Carslaw Publications, Glebe NSW, Australia, 1992.
5. 5.
T. Hibi and K. Matsuda, Quadratic Gröbner bases of twinned order polytopes, European J. Combin., 54 (2016), 187–192.
6. 6.
T. Hibi, K. Matsuda and A. Tsuchiya, Quadratic Gröbner bases arising from partially ordered sets, Math. Scand. 121 (2017), 19–25.
7. 7.
T. Hibi, K. Matsuda and A. Tsuchiya, Gorenstein Fano polytopes arising from order polytopes and chain polytopes. arXiv:1507.03221.
8. 8.
T. Hibi and A. Tsuchiya, Facets and volume of Gorenstein Fano polytopes, Math. Nachr. 290 (2017), 2619–2628.
9. 9.
H. Ohsugi and T. Hibi, Reverse lexicographic squarefree initial ideals and Gorenstein Fano polytopes, J. Commut. Alg., 10 (2018), 171–186.Google Scholar
10. 10.
A. Schrijver, “Theory of Linear and Integer Programing”, John Wiley & Sons, 1986.Google Scholar
11. 11.
R. P. Stanley, Two poset polytopes, Disc. Comput. Geom. 1 (1986), 9–23.