Automorphisms with eigenvalues in $S^1$ of a ${\mathbb Z}$-lattice with cyclic finite monodromy

TL;DR

This paper characterizes when automorphisms of certain cyclic ${f Z}$-lattices with eigenvalues on the unit circle are limited to powers of a fundamental automorphism, using cyclotomic polynomials and resultants.

Contribution

It provides necessary and sufficient conditions on the set of eigenvalues for automorphisms to be generated by the fundamental automorphism and its sign.

Findings

Identifies conditions on set M for automorphisms to be only ± powers of h_M

Uses cyclotomic polynomials and resultants in the proof

Applies results to automorphisms of Milnor lattices in singularity theory

Abstract

For any finite set $M\subset {\mathbb Z}_{\geq 1}$ of positive integers, there is up to isomorphism a unique ${\mathbb Z}$-lattice $H_M$ with a cyclic automorphism $h_M:H_M\to H_M$ whose eigenvalues are the unit roots with orders in $M$ and have multiplicity 1. The paper studies the automorphisms of the pair $(H_M,h_M)$ which have eigenvalues in $S^1$. The main result are necessary and sufficient conditions on the set $M$ such that the only such automorphisms are $\pm h_M^k,k\in{\mathbb Z}$. The proof uses resultants and cyclotomic polynomials. It is elementary, but involved. Special cases of the main result have been applied to the study of the automorphisms of Milnor lattices of isolated hypersurface singularities.