Six-dimensional exceptional quotient singularities

TL;DR

This paper classifies six-dimensional exceptional quotient singularities and proves the non-existence of seven-dimensional ones, highlighting the role of the Hall--Janko group in the six-dimensional case.

Contribution

It provides a complete classification of six-dimensional exceptional quotient singularities and identifies a specific group representation that yields such a singularity.

Findings

Six-dimensional exceptional quotient singularities are classified.

Seven-dimensional exceptional quotient singularities do not exist.

The Hall--Janko group representation produces an exceptional quotient singularity.

Abstract

We classify six-dimensional exceptional quotient singularities and show that seven-dimensional exceptional quotient singularities do not exist. Inter alia we prove that the irreducible six-dimensional projective representation of the sporadic simple Hall--Janko group gives rise to an exceptional quotient singularity.