Level algebras and $\boldsymbol{s}$-lecture hall polytopes

TL;DR

This paper classifies Gorenstein and level properties of $oldsymbol{s}$-lecture hall polytopes using combinatorial and geometric methods, enhancing understanding of their structure and providing explicit classifications.

Contribution

It offers the first concrete combinatorial and geometric classifications of Gorenstein and level $oldsymbol{s}$-lecture hall polytopes based on $oldsymbol{s}$-inversion sequences and tangent cones.

Findings

Classified Gorenstein and level properties via $oldsymbol{s}$-inversion sequences.

Provided geometric criteria for Gorenstein property in tangent cones.

Constructed infinite families of level $oldsymbol{s}$-lecture hall polytopes.

Abstract

Given a family of lattice polytopes, a common endeavor in Ehrhart theory is the classification of those polytopes in the family that are Gorenstein, or more generally level. In this article, we consider these questions for $\boldsymbol{s}$-lecture hall polytopes, which are a family of simplices arising from $\boldsymbol{s}$-lecture hall partitions. In particular, we provide concrete classifications for both of these properties purely in terms of $\boldsymbol{s}$-inversion sequences. Moreover, for a large subfamily of $\boldsymbol{s}$-lecture hall polytopes, we provide a more geometric classification of the Gorenstein property in terms of its tangent cones. We then show how one can use the classification of level $\boldsymbol{s}$-lecture hall polytopes to construct infinite families of level $\boldsymbol{s}$-lecture hall polytopes, and to describe level $\boldsymbol{s}$-lecture hall polytopes in small dimensions.