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

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.