On the characteristic polynomials of multiparameter pencils

In this paper we study the characteristic polynomial of multiparameter pencil z1A1+z2A2+⋯+zsAs. The main theorem states that a unitary representation of a finitely generated group contains a one-dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor. It follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible. The result is of kin to the Dedekind and Frobenius theorem on finite group determinant.