Self dual reflexive simplices with Eulerian polynomials

TL;DR

This paper introduces a new reflexive simplex with self-duality properties, whose delta-polynomial is the Eulerian polynomial, and demonstrates a compatible triangulation, advancing understanding of reflexive polytopes.

Contribution

The paper constructs a novel self-dual reflexive simplex and establishes its delta-polynomial as the Eulerian polynomial, along with a compatible triangulation.

Findings

The simplex $Q_n$ is reflexive and self-dual.

The delta-polynomial of $Q_n$ equals the Eulerian polynomial.

A regular, flag, unimodular triangulation of $Q_n$ exists.

Abstract

A lattice polytope $\mathcal{P}$ is called reflexive if its dual $\mathcal{P}^\vee$ is a lattice polytope. The property that $\mathcal{P}$ is unimodularly equivalent to $\mathcal{P}^\vee$ does not hold in general, and in fact there are few examples of such polytopes. In this note, we introduce a new reflexive simplex $Q_n$ which has this property. Additionally, we show that $\delta$-polynomalial of $Q_n$ is the Eulerian polynomial and show the existence of a regular, flag, unimodular triangulation.