A negative result of direct sum of integral symplectic matrices

In this paper we give a result on the symplectic direct sum of symplectic matrices. For any torsion of integral symplectic matrix X with cyclotomic polynomial as its characteristic polynomial, direct sum of k copies of X is not symplectic integral similar to direct sum of k copies of X−1, the inverse of X. The tool we use is symplectic group spaces constructed by pairs of ideal and element in domain Z[ζ] with certain conditions, where Z is the ring of integers and ζ is a root of a palindromic monic irreducible polynomial.