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arXiv 2004.02285math.RA
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关于矩阵代数的分次数

On the Number of Gradings on Matrix Algebras

Diogo Diniz, Daniel Pellegrino

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摘要：

确定了上块三角形矩阵代数上有限群的初等分次同构类的个数。作为结果，我们证明了，对于一个有限的阿贝尔群$G$，序列的同构类的$G$ -分次上的代数$M_{m}(\mathbb{F})$的$m\times m$矩阵的元素在一个字段$\mathbb{F}$特征$G$的数字$E(G,m)$。给出了代数闭域上块三角形矩阵代数上有限交换群分次同构类个数的一个公式，并对其性质作了适当的限制。最后，如果$G$是有限交换群，$\mathbb{F}$是代数闭域，$N(G,m)$是$G$ -分次在$M_{m}% (\mathbb{F})$上的同构类数，我们证明了$N(G,m)\sim\frac{1}{\left\vert G\right\vert !}m^{\left\vert G\right\vert -1}\sim E(G,m)$ .

英文摘要：

We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$ of isomorphism classes of elementary $G$-gradings on the algebra $M_{m}(\mathbb{F})$ of $m\times m$ matrices with entries in a field $\mathbb{F}$ characterizes $G$. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on its characteristic, is also provided. Finally, if $G$ is a finite abelian group, $\mathbb{F}$ is an algebraically closed field and $N(G,m)$ is the number of isomorphism classes of $G$-gradings on $M_{m}% (\mathbb{F})$ we prove that $N(G,m)\sim\frac{1}{\left\vert G\right\vert !}m^{\left\vert G\right\vert -1}\sim E(G,m)$.

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