On the kernel of the derivation operator

We state a necessary and sufficient condition for a symmetrized tensor to belong to the kernel of the derivation operator (the derivation operator, a generalization of the associated transformation introduced by M. Marcus [in: O. Shisha (Ed.), Inequalities II, Academic Press, New York (1970) p. 223], is a linear operator on the range of the symmetrizer) when the symmetrizer is associated to an absolutely irreducible character of Sm corresponding to the partition λ of m and the tensor is the image by this symmetrizer of the tensor product of a family of vectors of rank partition λ′.