A tropical analogue of the Hessian group

TL;DR

This paper explores a tropical analogue of the Hessian group, showing that the dihedral group d63d3 acts as the automorphism group on the tropical Hesse pencil through ultradiscretization.

Contribution

It introduces a tropical analogue of the Hessian group and identifies the automorphism group of the tropical Hesse pencil as d63d3, extending classical symmetry results to tropical geometry.

Findings

The group law on the Hesse pencil reduces to that on the tropical Hesse pencil.

The dihedral group d63d3 is the automorphism group of the tropical Hesse pencil.

Ultradiscretization connects classical and tropical automorphism groups.

Abstract

We investigate a tropical analogue of the Hessian group $G_{216}$, the group of linear automorphisms acting on the Hesse pencil. Through the procedure of ultradiscretization, the group law on the Hesse pencil reduces to that on the tropical Hesse pencil. We then show that the dihedral group $\mathcal{D}_3$ of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.