A short note on the image of multilinear graded polynomials on matrix algebras

Abstract: We investigate the subspaces obtained by images of multilinear graded polynomials evaluated on the matrix algebra endowed with the canonical $C_n$-grading, where $C_n$ denotes the cyclic group of order $n$. Moreover, this graded algebra admits a natural action of $C_n$ arising from a fine refinement of the grading. We prove that every $C_n$-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).

• The paper proves that every $C_n$-submodule of the trivial component of canonically graded $\mathrm{M}_n(\mathbb{F})$ occurs as the image of a multilinear graded polynomial when $\operatorname{char}\mathbb{F}\nmid n$.
• Using character-based polynomials and Galois descent, the authors realize irreducible components and their sums over arbitrary fields, disproving Centrone and de Mello’s conjecture.
• For $n=3$ over fields containing a primitive cube root of unity, the image of every multilinear graded polynomial studied is a vector subspace, while the general case remains open.

Context and motivation

The L'vov–Kaplansky problem asks whether the image of a multilinear polynomial evaluated on a matrix algebra is necessarily a vector subspace. Positive answers are known for $2\times 2$ matrices [Kanel-Belov, Malev, Rowen], with partial results for $3\times 3$ matrices, and the question has been studied for many classes of algebras including upper triangular matrices and Lie algebras. A graded variant of the problem was pursued by Centrone and de Mello, who studied multilinear polynomials in graded variables evaluated on $\mathrm{M}_n(\mathbb{Q})$ equipped with the canonical (Vasilovsky) $C_n$-grading. They described the linear span of such images over $\mathbb{Q}$ and conjectured that their description holds over an arbitrary base field. The paper under review, by da Silva and Yasumura, refutes that conjecture by proving a stronger structural statement: every $C_n$-submodule of the trivial component of the grading arises as the image of some multilinear polynomial in graded variables.

Setup

Throughout, $C_n=\langle\alpha\rangle$ is cyclic of order $n$, and $\mathbb{F}$ is a field with $\operatorname{char}\mathbb{F}=p\nmid n$. The canonical grading on $3\times 3$0 assigns to each $3\times 3$1 the span of matrix units $3\times 3$2 with $3\times 3$3. Identifying $3\times 3$4 with the cycle $3\times 3$5, the algebra becomes a $3\times 3$6-module via $3\times 3$7, an action compatible with the grading. Evaluations take place in the free $3\times 3$8-graded associative algebra $3\times 3$9, with $\mathrm{M}_n(\mathbb{Q})$0 denoting variables of homogeneous degree $\mathrm{M}_n(\mathbb{Q})$1.

For each character $\mathrm{M}_n(\mathbb{Q})$2, the authors introduce the element

$\mathrm{M}_n(\mathbb{Q})$3

which spans a one-dimensional irreducible $\mathrm{M}_n(\mathbb{Q})$4-submodule $\mathrm{M}_n(\mathbb{Q})$5 of the trivial component $\mathrm{M}_n(\mathbb{Q})$6, and the multilinear polynomial

$\mathrm{M}_n(\mathbb{Q})$7

Realizing irreducible submodules as images

The core technical observation is that, setting $\mathrm{M}_n(\mathbb{Q})$8 (indices modulo $\mathrm{M}_n(\mathbb{Q})$9), the polynomial $C_n$0 vanishes on all $C_n$1-tuples of the $C_n$2 except cyclic permutations of $C_n$3, on which it takes values in $C_n$4; since $C_n$5 is one-dimensional, this yields $C_n$6. Orthogonality is achieved via the elements $C_n$7: one has $C_n$8, where $C_n$9. Consequently, when $\mathbb{Q}$0 contains a primitive $\mathbb{Q}$1-th root of unity, any sum $\mathbb{Q}$2 has image exactly $\mathbb{Q}$3.

Dropping the roots-of-unity hypothesis

The general case is handled via Galois descent. Letting $\mathbb{Q}$4 be a splitting field of $\mathbb{Q}$5 over $\mathbb{Q}$6, the group algebra decomposes as $\mathbb{Q}$7 into simple components, and after scalar extension each primitive central idempotent $\mathbb{Q}$8 corresponds to a Galois orbit $\mathbb{Q}$9. The key point is that the orbit-summed polynomial

$C_n$0

has coefficients fixed by $C_n$1, hence lies in $C_n$2, and its image equals the submodule $C_n$3. Since every $C_n$4-submodule of the regular representation — identified here with $C_n$5 — is a direct sum of such components, summing the corresponding $C_n$6 gives the main theorem: every $C_n$7-submodule of $C_n$8 is the image of some multilinear polynomial in graded variables.

This immediately refutes Conjecture 1 of Centrone and de Mello: if all images were as conjectured, the attainable images would be restricted, whereas the theorem shows that arbitrary submodules — including proper subspaces of the spans they predicted — occur as images. The negative answer holds over any field whose characteristic does not divide $C_n$9.

Images as vector spaces: the $C_n=\langle\alpha\rangle$0 case

The authors also revisit the original L'vov-type question in the graded setting: is the image of a multilinear graded polynomial always a vector subspace? The polynomials $C_n=\langle\alpha\rangle$1, indexed by $C_n=\langle\alpha\rangle$2 and $C_n=\langle\alpha\rangle$3, form a basis of the degree-$C_n=\langle\alpha\rangle$4 multilinear polynomials in the $C_n=\langle\alpha\rangle$5. For $C_n=\langle\alpha\rangle$6 over a field containing a primitive cube root of unity, they prove that the image of any such polynomial is indeed a vector subspace. The proof splits into cases: evaluations at combinations of the $C_n=\langle\alpha\rangle$7 settle most coefficient configurations directly, while the remaining case — where coefficients $C_n=\langle\alpha\rangle$8 and $C_n=\langle\alpha\rangle$9 are both nonzero with vanishing sum — is handled by two explicit evaluations showing closure under scalar multiples and sums. Notably, this positive result is established only for $n$0; the general case remains open.

Limitations and open questions

Several restrictions should be noted. The main theorem concerns only submodules of the trivial component $n$1; whether analogous realization results hold for other homogeneous components is not addressed. The characteristic assumption $n$2 is essential to the semisimple representation theory used throughout, and the case $n$3 is left untreated. The positive answer on images being vector spaces requires $n$4 and a primitive cube root of unity in the base field; the authors explicitly leave open whether the image of a multilinear graded polynomial is a vector subspace in general.

Conclusion

This note establishes that the $n$5-module structure induced on the trivial component of the canonically graded matrix algebra is fully realized by images of multilinear graded polynomials: every submodule occurs as such an image. This yields a definitive negative answer to the 2023 conjecture of Centrone and de Mello and demonstrates that the graded version of the L'vov problem exhibits strictly richer behavior than its ungraded counterpart. The positive result for $n$6 suggests that the vector-subspace question in the graded setting may have a more delicate answer depending on $n$7 and the base field.