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

Abstract

Let p be a prime number. Denote by C(xm ) and C(xn ) the companion matrices of the polynomials xm and xn of positive degree over the field . Let ρ and σ be non-zero elements of an extension field K of . The Jordan form of the Kronecker product 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 × n matrices X over defined by and represented by the Kronecker sum matrix . 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 xL of S(m, n) and their multiplicities μ 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.