Volume, Facets and Dual Polytopes of Twinned Chain Polytopes

Let P and Q be finite partially ordered sets with |P | = |Q| = d, and C(P ) ⊂ R and C(Q) ⊂ R their chain polytopes. The twinned chain polytope of P and Q is the normal Gorenstein Fano polytope Γ(C(P ),−C(Q)) ⊂ R which is the convex hull of C(P )∪(−C(Q)). In this 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 characterize the facets of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the dual polytopes of twinned chain polytopes. introduction A convex polytope P ⊂ R is integral if all vertices belong to Z. An integral convex polytope P ⊂ R is normal if, for each integer N > 0 and for each a ∈ NP ∩ Z, there exist a1, . . . , aN ∈ P ∩ Z d such that a = a1 + · · · + aN , where NP = {Nα | α ∈ P}. Furthermore, an integral convex polytope P ⊂ R is Fano if the origin of R is a unique integer point belonging to the interior of P. A Fano polytope P ⊂ R is Gorenstein if its dual polytope P := {x ∈ R | 〈x,y〉 ≤ 1 for all y ∈ P} is integral as well. A Gorenstein Fano polytope is also said to be a reflexive polytope. In resent years, the study of Gorenstein Fano polytopes has been more vigorous. It is known that Gorenstein Fano polytopes correspond to Gorenstein toric Fano varieties, and they are related with mirror symmetry (see, e.g., [1, 2]). We recall some terminologies of partially ordered sets. Let P = {p1, . . . , pd} be a partially ordered set. A linear extension of P is a permutation σ = i1i2 · · · id of [d] = {1, 2, . . . , d} which satisfies ia < ib if pia < pib in P . A subset I of P is called a poset ideal of P if pi ∈ I and pj ∈ P together with pj ≤ pi guarantee pj ∈ I. Note that the empty set ∅ and P itself are poset ideals of P . Let J (P ) denote the set of poset ideals of P . A subset A of P is called an antichain of P if pi and pj belonging to A with i 6= j are incomparable. In particular, the empty set ∅ and each 1-elemant subsets {pj} are antichains of P . Let A(P ) denote the set of antichains of P . For 2010 Mathematics Subject Classification. 52B05, 52B20.