Vasiliki Petrotou: The Hibi-Oshugi Conjecture for IDP and Gorenstein Polytopes

Abstract: A lattice polytope P is a convex polytope whose vertices all have integer coordinates. Given a field k, one can associate to P its Ehrhart ring k[P], a graded commutative k-algebra whose Hilbert function counts the number of lattice points in the dilations of P. The polytope P is called Gorenstein if k[P] is Gorenstein, and it satisfies the Integer Decomposition Property (IDP) if k[P] is generated in degree 1. The numerator of the Hilbert series of k[P] is the h^{\star}-polynomial of P, whose coefficients form the h^{\star} vector. In 2006 Hibi and Ohsugi conjectured that for IDP Gorenstein polytopes, the h^{\star}-vector is unimodal, meaning its entries never strictly increase after the first time they strictly decrease. In this talk I will discuss a recent proof of this conjecture (joint work with Karim Adiprasito, Stavros Papadakis and Johanna Steinmeyer) and sketch the algebraic ideas behind it.