Constructing the identities and the central identities of degree < 9 of the n x n matrices

Abstract

<p>In this paper, we present an algorithm that can be used to compute an independent generating set for the multilinear identities or multilinear central identities of the nx n matrices M[subscript]n([phi]) over a field [phi], where [phi] is of characteristic zero or a large enough prime. Furthermore, we prove that every identity or central identity is implied by the set of multilinear identities or central identities which the algorithm produces. Therefore, the procedure creates all the identities and central identities of degree n. Then we use this algorithm to construct all the multilinear identities and multilinear central identities of degree n<9 for M[subscript]3([phi]). The results of the procedure are given in chapter 3. There are no identities or central identities of degree n<6. The only multilinear identity of degree 6 is the standard identity of degree 6 which is given by (3.3). The multilinear identities (3.4)-(3.9) form an independent generating set for all the multilinear identities of degree 7. All of these identities are consequences of the standard identity of degree 6. Finally, the identities (3.10)-(3.11) together with the central identities (3.12)-(3.15) form an independent generating set for all of the multilinear central identities of degree 8.</p>