Gorenstein properties and integer decomposition properties of lecture hall polytopes

TL;DR

This paper investigates the Gorenstein and integer decomposition properties of lecture hall polytopes, establishing conditions under which these properties hold and providing constructions for Gorenstein/IDP polytopes.

Contribution

It proves that monotonic s-lecture hall polytopes satisfy the IDP and characterizes conditions for Gorenstein, Fano, and reflexive properties, including a construction method.

Findings

Monotonic s-lecture hall polytopes satisfy the IDP.

Necessary and sufficient conditions for Gorenstein, Fano, and reflexive properties.

A construction method for Gorenstein/IDP lecture hall polytopes.

Abstract

Though much is known about ${\bf s}$-lecture hall polytopes, there are still many unanswered questions. In this paper, we show that ${\bf s}$-lecture hall polytopes satisfy the integer decomposition property (IDP) in the case of monotonic ${\bf s}$-sequences. Given restrictions on a monotonic ${\bf s}$-sequence, we discuss necessary and sufficient conditions for the Fano, reflexive and Gorenstein properties. Additionally, we give a construction for producing Gorenstein/IDP lecture hall polytopes.