Some algebra with formal matrices

Abstract

The aim of this paper is 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 an ideal 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 k[S _m] of the symmetric group on the m ^th tensor product of E=k ⁿ. 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, product or traces of the well-known Cayley-Hamilton identity. This leads to an algorithm for reducing to a canonical form modulo the ideal of identities of matrices in the free algebra.