The combinatorics of weight systems and characteristic polynomials of isolated quasihomogeneous singularities

TL;DR

This paper proves a key conjecture about the characteristic polynomial of isolated quasihomogeneous singularities and explores related combinatorial problems, impacting understanding of monodromy and automorphisms in singularity theory.

Contribution

It confirms a major conjecture on the characteristic polynomial and advances combinatorial methods in the study of quasihomogeneous singularities.

Findings

Proof of the conjecture on the characteristic polynomial

Solutions to several combinatorial problems related to singularities

Implications for the automorphism group of the Milnor lattice

Abstract

A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the characteristic polynomial. Here the conjecture is proved, and some of the other problems are solved, too. In the cases where also an old conjecture of Orlik on the integral monodromy holds, this has implications on the automorphism group of the Milnor lattice. The combinatorics used in the proof of the conjecture consists of tuples of orders on sets $\{0,1,...,n\}$ with special properties and may be of independent interest.