Level algebras and s-lecture hall polytopes

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 s-lecture hall polytopes, which are a family of simplices arising from s-lecture hall partitions. In particular, we provide concrete classifications for both of these properties purely in terms of sinversion sequences. Moreover, for a large subfamily of 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 s-lecture hall polytopes to construct infinite families of level s-lecture hall polytopes, and to describe level s-lecture hall polytopes in small dimensions.