Commutation relations and Vandermonde determinants

We consider a certain decomposition of the matrix algebra Mn(F)Mn(F), where FF is a field. The commutation relations of that decomposition yield an n2×n2n2×n2 matrix MMn(F)MMn(F), which determines the multilinear polynomial identities of Mn(F)Mn(F). Thus if char(F)=0, the matrix MMn(F)MMn(F) determines the polynomial identities of Mn(F)Mn(F). We show that MMn(F)MMn(F) is very close to the tensor product of two n×nn×n Vandermonde matrices. In particular this allows us to evaluate the determinant of MMn(F)MMn(F).