Automorphisms with eigenvalues in S¹ of a Z-lattice with cyclic finite monodromy

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For any finite set M⊂ Z≥ ₁ of positive integers, there is up to isomorphism a unique Z-lattice Hₘ with a cyclic automorphism hₘ:Hₘ→ Hₘ whose eigenvalues are the unit roots with orders in M and have multiplicity 1. The paper studies the automorphisms of the pair (Hₘ,hₘ) which have eigenvalues in S¹. The main result are necessary and sufficient conditions on the set M such that the only such automorphisms are ± hₘᵏ,k∈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.