The general question
concerning the structure of subspaces of a symmetry class of tensors in which every
nonzero element has an irreducible representation as a sum of decomposable (or
pure) elements of a given length is as yet largely unanswered. This problem
relates to the problem of characterizing the linear transformations on such a
symmetry class which map the set of tensors of “irreducible length” k into itself;
i.e., preserves the rank k of the tensors. Another related problem is: “Is it
possible to obtain algebraic relations involving the components of a tensor
which imply it has rank (“Irreducible length”) k, for any positive integer
k”?
