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arXiv · 2608.16768

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

Italo Bauer Rodrigues da Silva·Felipe Yukihide Yasumura
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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).

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Italo Bauer Rodrigues da Silva, Felipe Yukihide Yasumura. 2026-08-17. A short note on the image of multilinear graded polynomials on matrix algebras. https://arxiv.org/abs/2608.16768

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