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-MINIMUM SPANNING LENGTHS AND AN EXTENSION TO BURNSIDE’S THEOREM ON IRREDUCIBILITY
Bulletin of the Australian Mathematical Society ( IF 0.7 ) Pub Date : 2020-12-02 , DOI: 10.1017/s0004972720001227 W. E. LONGSTAFF
Bulletin of the Australian Mathematical Society ( IF 0.7 ) Pub Date : 2020-12-02 , DOI: 10.1017/s0004972720001227 W. E. LONGSTAFF
We introduce the $\textbf{h}$ -minimum spanning length of a family ${\mathcal A}$ of $n\times n$ matrices over a field $\mathbb F$ , where $\textbf{h}$ is a p -tuple of positive integers, each no more than n . For an algebraically closed field $\mathbb F$ , Burnside’s theorem on irreducibility is essentially that the $(n,n,\ldots ,n)$ -minimum spanning length of ${\mathcal A}$ exists if ${\mathcal A}$ is irreducible. We show that the $\textbf{h}$ -minimum spanning length of ${\mathcal A}$ exists for every $\textbf{h}=(h_1,h_2,\ldots , h_p)$ if ${\mathcal A}$ is an irreducible family of invertible matrices with at least three elements. The $(1,1, \ldots ,1)$ -minimum spanning length is at most $4n\log _{2} 2n+8n-3$ . Several examples are given, including one giving a complete calculation of the $(p,q)$ -minimum spanning length of the ordered pair $(J^*,J)$ , where J is the Jordan matrix.
中文翻译:
-Burnside 不可约性定理的最小跨度和扩展
我们介绍 $\textbf{h}$ - 一个家庭的最小跨越长度 ${\数学A}$ 的 $n\次 n$ 域上的矩阵 $\mathbb F$ , 在哪里 $\textbf{h}$ 是一个p - 正整数元组,每个不超过n . 对于代数闭域 $\mathbb F$ , Burnside 关于不可约性的定理本质上是 $(n,n,\ldots ,n)$ - 最小跨越长度 ${\数学A}$ 如果存在 ${\数学A}$ 是不可约的。我们证明了 $\textbf{h}$ - 最小跨越长度 ${\数学A}$ 存在于每个 $\textbf{h}=(h_1,h_2,\ldots , h_p)$ 如果 ${\数学A}$ 是具有至少三个元素的不可约的可逆矩阵族。这 $(1,1, \ldots ,1)$ - 最小跨越长度最多为 $4n\log _{2} 2n+8n-3$ . 给出了几个例子,其中一个给出了一个完整的计算 $(p,q)$ - 有序对的最小跨越长度 $(J^*,J)$ , 在哪里Ĵ 是乔丹矩阵。
更新日期:2020-12-02
中文翻译:
-Burnside 不可约性定理的最小跨度和扩展
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