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

Final version, to appear in Electronic Journal of Combinatorics

References in corpus (3)

• Ehrhart Theory of Spanning Lattice Polytopes
• Generating functions and triangulations for lecture hall cones
• Gorenstein properties and integer decomposition properties of lecture hall polytopes