Some Algebraic Properties of Lecture Hall Polytopes

TL;DR

This paper explores algebraic and geometric properties of lecture hall polytopes, proving conjectures about their integer decomposition, triangulations, and Gr"obner bases, thus advancing understanding of their combinatorial structure.

Contribution

It proves that all s-lecture hall order polytopes satisfy the integer decomposition property and establishes quadratic Gr"obner bases for certain lecture hall simplices.

Findings

All s-lecture hall order polytopes satisfy the integer decomposition property.

Families of s-lecture hall simplices admit quadratic Gr"obner bases with square-free initial ideals.

Certain s-lecture hall simplices have regular, unimodular triangulations.

Abstract

In this note, we investigate some of the fundamental algebraic and geometric properties of $s$-lecture hall simplices and their generalizations. We show that all $s$-lecture hall order polytopes, which simultaneously generalize $s$-lecture hall simplices and order polytopes, satisfy a property which implies the integer decomposition property. This answers one conjecture of Hibi, Olsen and Tsuchiya. By relating $s$-lecture hall polytopes to alcoved polytopes, we then use this property to show that families of $s$-lecture hall simplices admit a quadratic Gr\"obner basis with a square-free initial ideal. Consequently, we find that all $s$-lecture hall simplices for which the first order difference sequence of $s$ is a $0,1$-sequence have a regular and unimodular triangulation. This answers a second conjecture of Hibi, Olsen and Tsuchiya, and it gives a partial answer to a conjecture of Beck, Braun, K\"oppe, Savage and Zafeirakopoulos.