Symmetric cubic surfaces and G_2 character varieties

TL;DR

This paper explores a special symmetric family of cubic surfaces and demonstrates their connection to G_2 character varieties, linking algebraic geometry, group theory, and automorphisms.

Contribution

It identifies a symmetric subfamily of cubic surfaces as G_2 character varieties and relates them to Spin(8) automorphisms and finite groups.

Findings

Symmetric cubic surfaces correspond to G_2 character varieties.

Connection established between symmetric surfaces and Spin(8) automorphisms.

Example involving a finite simple group inside G_2 analyzed.

Abstract

We will consider a two dimensional "symmetric" subfamily of the four dimensional family of Fricke cubic surfaces. The main result is that such symmetric cubic surfaces arise as character varieties for the exceptional group of type G_2. Further, this symmetric family will be related to the fixed points of the triality automorphism of Spin(8), and an example involving the finite simple group of order 6048 inside G_2 will be considered.