Before revision:  We consider the problem of generating an  n-cross field using a higher-order Q-tensor theory that is constructed out of tensored projection matrices.
After revision:  It was recently by other authors that 3-cross fields can be embedded into the set of symmetric 4th-order tensors. Another concurrent work further develops a relaxation of this tensor field via a certain set of varieties. In this paper, we consider the problem of generating an n-cross field using a higher-order Q-tensor theory that is constructed out of tensored projection matrices.