A negative result of direct sum of integral symplectic matrices

Abstract

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.