Level Algebras and $\mathbf {s}$-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 ${\boldsymbol s}$-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 ${\boldsymbol s}$-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.

DIVIDED POWER ALGEBRAS OVER AN OPERAD

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group and the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.