On Jordan bases for the tensor product and Kronecker sum and their elementary divisors over fields of prime characteristic

Let p be a prime number. Denote by C(x(m)) and C(x(n)) the companion matrices of the polynomials x(m) and x(n) of positive degree over the field Z(p). Let p and sigma be non-zero elements of an extension field K of Z(p). The Jordan form of the Kronecker product (pI +C(x(m)))circle times (sigma I+C(x(n))) of invertible Jordan block matrices over K is determined via an equivalent study of the nilpotent transformation S( m, n) of the vector space of m x n matrices X over Zp defined by (X)S(m, n) = C(x(m))X-T+XC(x(n)) and represented by the Kronecker sum matrix C(x(m))circle times I+I circle times C(x(n)). Using the p-adic expansions of m and n, an inductive method of constructing a Jordan basis for S( m, n) is described; this method is direct and based on classical formulae. The elementary divisors x L of S( m, n) and their multiplicities mu are specified in terms of these p-adic expansions, thus allowing computations in the representation algebra of a finite cyclic p-group to be carried out more readily than previously.