On the $K3$ surface with $\mathfrak{S}_4 \times \mathfrak{S}_4$ action

TL;DR

This paper provides explicit descriptions and characterizations of a specific K3 surface with an automorphism group of _4 imes _4, including an isomorphism to Schur's quartic and analysis of its automorphism groups.

Contribution

It offers three explicit characterizations of the K3 surface with _4 imes _4 symmetry and constructs an explicit isomorphism to Schur's quartic, expanding understanding of automorphism groups.

Findings

Explicit descriptions of the K3 surface with _4 imes _4 action

Construction of an explicit isomorphism to Schur's quartic

Calculation of intersection of polarization-preserving automorphism groups

Abstract

By a lattice theoretic approach, Brandhorst--Hashimoto has made the list of K3 surfaces with finite groups of automorphisms which properly contain a maximal symplectic automorphism group. We give $3$ different explicit descriptions to the $K3$ surface with an action of $\mathfrak{S}_4\times \mathfrak{S}_4$, with various characterizations, and construct an explicit isomorphism to the Schur's quartic. We also calculate the intersection of the two polarization-preserving finite automorphism groups.