We investigate the subspaces obtained by images of multilinear graded polynomials evaluated on the matrix algebra endowed with the canonical Cn-grading, where Cn denotes the cyclic group of order n. Moreover, this graded algebra admits a natural action of Cn
Nearby in the stack
arising from a fine refinement of the grading. We prove that every
Cn
-submodule of the trivial component of the grading is the image of some multilinear polynomial in graded variables. In particular, we answer in the negative a recent conjecture posed by T. de Castilho and L. Centrone (2023).