Commutation relations and Vandermonde determinants (Q1024325)

The matrix algebra \(A=M_n(F)\) of \(n \times n\) matrices over a field \(F\) with regard to the regular decomposition \(A=A_1 \oplus \dots\oplus A_R\) as a direct sum is analyzed. The commutation relations of the given decomposition give the \(n^2 \times n^2\) matrix \(M^{M_n(F)}\). It is shown that this matrix \(M^{M_n(F)}\), which determines the multilinear polynomial identities of \(M_n(F)\), is very close to the tensor product of two generic \(n \times n\) Vandermonde matrices and is obtained from such a tensor product by a natural row permutation and by a natural substitution. Thus it is possible to evaluate the determinant of \(M^{M_n(F)}\).