h-Vectors of Gorenstein polytopes

Abstract

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a "long" regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP , c ∈ N, can be written as the sum of c lattice points in P . (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.

Lipschitz polytopes of posets and permutation statistics

“…Stanley [18] noted that, in the case that an n-dimensional polytope P is r-Gorenstein, one has h * i (P) = h * n+1−r−i (P) for all r. Bruns-Römer [2] showed that if an r-Gorenstein polytope has a regular and unimodular triangulation, then the h * -vector is unimodal. For Lipschitz polytopes, such a triangulation is vouched for by Theorem 3.1 and with Proposition 4.2, we get the following.…”

Facets of the r-Stable (n, k)-Hypersimplex

“…In [3], it is shown that a Gorenstein integer polytope with a regular unimodular triangulation has a unimodal δ -vector. In [2], it is shown that ∆ stab(r) n, k has a regular unimodular triangulation.…”

On the facets of the secondary polytope

“…The G-property is related to the notion of Gorenstein polytopes [8], Gorenstein simplicial complexes, and Gorenstein rings [7,28] We call a sum w 1 + w 2 of two weight functions of a point configuration A coherent if…”