Commutation relations and Vandermonde determinants

We consider a certain decomposition of the matrix algebra M n ( F ) , where F is a field. The commutation relations of that decomposition yield an n 2 × n 2 matrix M M n ( F ) , which determines the multilinear polynomial identities of M n ( F ) . Thus if char ( F ) = 0 , the matrix M M n ( F ) determines the polynomial identities of M n ( F ) . We show that M M n ( F ) is very close to the tensor product of two n × n Vandermonde matrices. In particular this allows us to evaluate the determinant of M M n ( F ) .