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

Sven Balnojan, Claus Hertling

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.

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

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.

Paper Structure

This paper contains 8 sections, 36 theorems, 178 equations, 4 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Unimodular bilinear lattices with triangular bases
  3. Review of results on the simply laced root lattices
  4. Tubular elliptic root systems and their Coxeter elements
  5. Isolated hypersurface singularities
  6. The positive definite cases
  7. The positive semidefinite cases
  8. Conjectures for the indefinite cases

Key Result

Theorem 1.2

Fix $n\in{\mathbb N}$, a singularity $f=f(z_0,...,z_m)$ with Milnor number $n$, and a distinguished matrix $S\in T_n({\mathbb Z})$ of $f$. The matrix $S$ and its Coxeter-Dynkin diagram ${\rm CDD}(S)$ have the following properties. The matrix $S^{-1}S^t$ arises as a monodromy matrix. (a) (Monodromy t

Figures (4)

  • Figure 3.1: Dynkin diagrams of the ADE root lattices
  • Figure 4.1: Dynkin diagrams of the tubular elliptic root systems
  • Figure 4.2: Extended Dynkin diagrams
  • Figure 4.3: Coefficients of the generating relation

Theorems & Definitions (58)

  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['t6.2']} (a)
  • Conjecture 1.4
  • Theorem 1.5
  • Definition 2.1: HL24
  • Definition 2.3: HL24
  • Lemma 2.4
  • Lemma 2.6
  • Definition 2.7: HL24
  • Lemma 2.9
  • ...and 48 more