Generating functions and triangulations for lecture hall cones

TL;DR

This paper explores the structure of lecture hall cones, revealing their connections to reflexive simplices, Eulerian polynomials, and triangulations, and provides explicit descriptions and conjectures about their properties.

Contribution

It establishes isomorphisms between lecture hall cones and cones over reflexive simplices, describes their Hilbert basis, and proves the existence of certain triangulations.

Findings

Lecture hall cones are isomorphic to cones over reflexive simplices.

The Ehrhart h*-polynomial of these simplices is given by Eulerian polynomials.

Lecture hall cones admit regular, flag, unimodular triangulations.

Abstract

We investigate the arithmetic-geometric structure of the lecture hall cone \[ L_n \ := \ \left\{\lambda\in \mathbb{R}^n: \, 0\leq \frac{\lambda_1}{1}\leq \frac{\lambda_2}{2}\leq \frac{\lambda_3}{3}\leq \cdots \leq \frac{\lambda_n}{n}\right\} . \] We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of $L_n$, including connections between enumerative and algebraic properties of $L_n$ and cones over unit cubes.