Characterization of distinguished matrices of isolated hypersurface singularities through their spectral numbers

TL;DR

This paper characterizes distinguished matrices of isolated hypersurface singularities using spectral numbers, focusing on their variance and the role of ADE root lattices, with results applicable to positive definite and semidefinite cases.

Contribution

It introduces a spectral number variance criterion to identify distinguished matrices of hypersurface singularities, utilizing ADE root lattices and Weyl group representations.

Findings

Successful characterization in positive definite cases

Extension to positive semidefinite cases

Use of ADE root lattices and Weyl group elements

Abstract

Isolated hypersurface singularities come equipped with distinguished bases of their Milnor lattices and with upper triangular integral matrices, which are called here distinguished matrices. These matrices form an orbit of a braid group and a sign change group. This paper proposes to characterize the distinguished matrices of singularities within all upper triangular integral matrices in terms of the variance of certain spectral numbers. It succeeds in the positive definite and the positive semidefinite cases. The ADE root lattices are crucial. In the semidefinite cases, results on non-reduced presentations of Weyl group elements are used.