The Integer Decomposition Property and Weighted Projective Space Simplices

TL;DR

This paper classifies reflexive weighted projective space simplices with the integer decomposition property for up to three distinct non-unit weights, introduces reflexive stabilizations, and explores their properties related to unimodality and the integer decomposition property.

Contribution

It provides a complete classification for simplices with up to three non-unit weights, introduces the concept of reflexive stabilization, and analyzes their algebraic and combinatorial properties.

Findings

Complete classification for up to three non-unit weights.

Reflexive stabilizations with m ≥ 2 lack the integer decomposition property.

Large reflexive stabilizations have non-unimodal Ehrhart h*-polynomials.

Abstract

Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice simplices defining weighted projective spaces. We investigate the question of when a reflexive weighted projective space simplex has the integer decomposition property. We provide a complete classification of reflexive weighted projective space simplices having the integer decomposition property for the case when there are at most three distinct non-unit weights, and conjecture a general classification for an arbitrary number of distinct non-unit weights. Further, for any weighted projective space simplex and $m\geq 1$, we define the $m$-th reflexive stabilization, a reflexive weighted projective space simplex. We prove that when $m$ is $2$ or greater, reflexive stabilizations do not have the integer decomposition property. We also prove that the Ehrhart $h^\ast$-polynomial of any sufficiently large reflexive stabilization is not unimodal and has only $1$ and $2$ as coefficients. We use this construction to generate interesting examples of reflexive weighted projective space simplices that are near the boundary of both $h^*$-unimodality and the integer decomposition property.