Computable identities in the algebra of formal matrices

AbstractIn this paper, we want to give an explicit description of identities satisfied by matrices n × n over a field k of characteristic 0, in order to be able to compute with formal matrices (“forgetting” their representations with coefficients). We introduce a universal free algebra, where all formal manipulations are made. Using classical properties of identities in an algebra with trace, we reduce our problem to the study of identities among multilinear traces. These are closely linked with the action of the algebra of the symmetric group k[Sm] on the mth tensor product of E = Kn. By proving a theorem about the kernel of this action and its effective version, we can decompose all identities of matrices in an explicit way as linear combinations, substitutions, products or traces of the well-known Cayley-Hamilton identity. This leads to an algorithm of reduction to a canonical form, modulo the ideal of identities of matrices, in the free algebra. Moreover, this set of identities is in fact a prime ideal and a natural extension of the quotient by this ideal gives us a skew field, which is also the fraction ring of the algebra of generic matrices