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

Abstract

Given a family of lattice polytopes, a common question in Ehrhart theory is classifying which polytopes in the family are Gorenstein. A less common question is classifying which polytopes in the family admit level semigroup algebras, a generalization of Gorenstein algebras. In this article, we consider these questions for s-lecture hall polytopes, which are a family of simplices arising from lecture hall partitions. We provide a complete characterization of the Gorenstein property for lecture hall polytopes. We provide a different, more geometric characterization of the Gorenstein property which applies to a large subfamily of lecture hall polytopes. Additionally, we also give a complete characterization for the level property in terms of s-inversion sequences and demonstrate some consequences of the classification result. We conclude briefly with some potential future work and extensions of these results.Contents 14 5. Concluding remarks and future directions 17 References 18

Derangements, Ehrhart theory, and local h-polynomials

“…In relation to unimodality conjectures for h * -polynomials for reflexive polytopes, [11,40] asked which families of well-studied lattice polytopes do (or do not) admit a box unimodal triangulation. A second main contribution of this paper is to prove that the well-studied family of s-lecture hall simplices [7,8,13,29,36,37,38] all have real-rooted, and thus unimodal, local h * -polynomials (see Section 3). We then apply this result to prove that a family of lattice polytopes simultaneously generalizing s-lecture hall simplices and order polytopes [43], called the s-lecture hall order polytopes [13], admit a box unimodal triangulation.…”

Polyhedral geometry for lecture hall partitions

“…The Gorenstein property was studied in limited cases by Hibi Tuschiya, and the author in [22]. These results have been fully generalized by Kohl and the author in [24].…”

“…, n + 1), Hibi, Tsuchiya, and the author [21] show that P ; z) is the usual Eulerian polynomial A n+1 (z). In addition, Kohl and the author [24] are able to also provide a characterization of the level property in terms of s-inversion sequences. The proof of this theorem relies heavily on the fact that P (s) n is a simplex and studying the additive properties of the lattice points in Π P (s) n using the bijection that the lattice points at height i correspond to inversion sequences with i inversions.…”

“…Choices for grading which have been used in the literature include the standard "height" grading used in the polytope case, as well as the difference of last two coordinates which was used heavily in the case of C (1,2,...,n) n . After choosing a grading, it seems plausible that one may be able to adapt the strategy of Kohl and the author in the P (s) n case [24], though one would need to understand the Hilbert basis for any given cone considered.…”

Cayley sums and Minkowski sums of lattice polytopes

“…In particular, if P is level of index r and | int(rP) ∩Z N | = 1, then P is called Gorenstein of index r. The Gorenstein polytopes give important examples in combinatorial commutative algebra, mirror symmetry and tropical geometry (for details we refer to [1,12]). On the other hand, the level property is a generalization of the Gorenstein property and it has only fairly recently been examined for certain classes polytopes (e.g., [9,10,11]). Now, we see a connection between level polytopes and commutative algebras.…”