Automorphisms of $P_8$ singularities and the complex crystallographic groups

TL;DR

This paper classifies smoothable automorphisms of $P_8$ singularities related to complex crystallographic groups, identifying their monodromy groups as complex affine reflection groups, thus advancing the understanding of symmetries in singularity theory.

Contribution

It completes the classification of automorphisms of $P_8$ singularities and links their monodromy groups to known complex affine reflection groups.

Findings

Classification of smoothable automorphisms of $P_8$ singularities.

Identification of monodromy groups as complex affine reflection groups.

Extension of previous symmetry studies in singularity theory.

Abstract

The paper completes the study of symmetries of parabolic function singularities with relation to complex crystallographic groups that was started in \cite{GM,X9}. We classify smoothable automorphisms of $P_8$ singularities which split the kernel of the intersection form on the second homology. For such automorphisms, the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of complex affine reflection groups tabled in \cite{P}.