The h-Vector of a Gorenstein Toric Ring of a Compressed Polytope

A compressed polytope is an integral convex polytope all of whose pulling triangulations are unimodular. A (q − 1)-simplex Σ each of whose vertices is a vertex of a convex polytope P is said to be a special simplex in P if each facet of P contains exactly q − 1 of the vertices of Σ. It will be proved that there is a special simplex in a compressed polytope P if (and only if) its toric ring K[P] is Gorenstein. In consequence it follows that the h-vector of a Gorenstein toric ring K[P] is unimodal if P is compressed. A compressed polytope [10, p. 337] is an integral convex polytope all of whose “pulling triangulations” are unimodular. (Recall that an integral convex polytope is an convex polytope each of whose vertices has integer coordinates.) A typical example of compressed polytopes is the Birkhoff polytopes [10, Example 2.4 (b)]. Later, in [6], a large class of compressed polytopes including the Birkhoff polytopes is presented. Recently, Seth Sullivant [12] proved a surprising result that the class given in [6] does essentially contain all compressed polytopes. the electronic journal of combinatorics 11(2) (2005), #N4 1 Let P ⊂ R be an integral convex polytope. Let K be a field and K[x,x−1, t] = K[x1, x −1 1 , . . . , xn, x −1 n , t] the Laurent polynomial ring in n+1 variables over K. The toric ring of P is the subalgebra K[P] of K[x,x−1, t] which is generated by those monomials xt = x1 1 · · ·xan n t such that a = (a1, . . . , an) belongs to P ⋂ Z. We will regard K[P] as a homogeneous algebra [2, p. 147] by setting each deg xt = 1 and write F (K[P], λ) for its Hilbert series. One has F (K[P], λ) = (h0+h1λ+ · · ·+hsλ)/(1−λ), where each hi ∈ Z with hs 6= 0 and where d is the dimension of P. The sequence h(K[P]) = (h0, h1, . . . , hs) is called the h-vector of K[P]. If the toric ring K[P] is normal, then K[P] is Cohen– Macaulay. If K[P] is Cohen–Macaulay, then the h-vector of K[P] is nonnegative, i.e., each hi ≥ 0. Moreover, if K[P] is Gorenstein, then the h-vector of K[P] is symmetric, i.e., hi = hs−i for all i. A well-known conjecture is that the h-vector (h0, h1, . . . , hs) of a Gorenstein toric ring is unimodal, i.e., h0 ≤ h1 ≤ · · · ≤ h[s/2]. One of the effective techniques to show that (h0, h1, . . . , hs) is unimodal is to find a simplicial complex polytope of dimension s − 1 whose h-vector [11, p. 75] coincides with (h0, h1, . . . , hs) (Stanley [9]). In fact, Reiner and Welker [8] succeeded in showing that the h-vector of a Gorenstein toric ring arising from a finite distributive lattice (see, e.g., [4]) is equal to the h-vector of a simplicial convex polytope. Christos Athanasiadis [1] introduced the concept of a “special simplex” in a convex polytope. Let P ⊂ R be a convex polytope. A (q − 1)-simplex Σ each of whose vertices is a vertex of P is said to be a special simplex in P if each facet (maximal face) of P contains exactly q − 1 of the vertices of Σ. It turns out [1, Theorem 3.5] that if P is compressed and if there is a special simplex in P, then the h-vector of K[P] is equal to the h-vector of a simplicial convex polytope. In particular, if P is compressed and if there is a special simplex in P, then K[P] is Gorenstein whose h-vector is unimodal. Examples for which [1, Theorem 3.5] can be applied include (i) toric rings of the Birkhoff polytopes ([1, Example 3.1]), (ii) Gorenstein toric rings arising from finite distributive lattices ([1, Example 3.2]), and (iii) Gorenstein toric rings arising from stable polytopes of perfect graphs ([7, Theorem 3.1 (b)]). In the present paper we prove that there is a special simplex in a compressed polytope P if (and only if) its toric ring K[P] is Gorenstein. Theorem 0.1 Let P be a compressed polytope. Then there exists a special simplex in P if (and only if) its toric ring K[P] is Gorenstein. Proof. It follows from [12, Theorem 2.4] that every compressed polytope P is lattice isomorphic to an integral convex polytope of the form Cn ⋂ L, where Cn ⊂ R is the n-dimensional unit hypercube and where L is an affine subspace of R. Without loss of generality, one can assume that L ⋂ (Cn \ ∂Cn) 6= ∅, where ∂Cn is the boundary of Cn. In other words, dimP = dim L. Let P = Cn ⋂ L with d = dimP. Thus L is the intersection of n − d hyperplanes in R, say a11x1 + · · ·+ a1dxd + xd+1 = b1 a21x1 + · · ·+ a2dxd + xd+2 = b2 the electronic journal of combinatorics 11(2) (2005), #N4 2 · · · an−d,1x1 + · · · + an−d,dxd + xn = bn−d, where aij , bi ∈ Q for all i and j. Since P possesses the integer decomposition property [6, p. 2544], its toric ring coincides with the Ehrhart ring [5, p. 97] of P. Hence the criterion [3, Corollary 1.2] can be applied for K[P]. To state the criterion [3, Corollary 1.2], let δ > 0 denote the smallest integer for which δ(P \ ∂P) ⋂ Z 6= ∅, where δ(P \ ∂P) = {δα : α ∈ P \ ∂P}, and (c1, . . . , cn) ∈ δ(P \ ∂P) ⋂ Z. Write Q ⊂ R for the convex polytope defined by the inequalities 0 ≤ xi ≤ 1, 1 ≤ i ≤ d