关于矩阵代数上多重线性分次多项式像的短注
A short note on the image of multilinear graded polynomials on matrix algebras
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中文总结 AI 辅助
本文研究典范$C_n$分次矩阵代数上多重线性分次多项式的像子空间，证明分次平凡分量的每个$C_n$子模均为对应多项式的像，否定了de Castilho与Centrone2023年的猜想。
中文摘要 AI 辅助
我们研究在配备典范$C_n$分次的矩阵代数上计算多重线性分次多项式所得的子空间，其中$C_n$表示$n$阶循环群。此外，该分次代数具有由分次的精细细化产生的自然$C_n$作用。我们证明，分次平凡分量的每个$C_n$子模都是某个分次变量多重线性多项式的像。特别地，我们否定了T. de Castilho和L. Centrone（2023）提出的近期猜想。
英文摘要
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).
