We introduce the notion of the slot length of a family of matrices over an arbitrary field ${\mathbb {F}}$ . Using this definition it is shown that, if $n\ge 5$ and A and B are $n\times n$ complex matrices with A unicellular and the pair $\{A,B\}$ irreducible, the slot length s of $\{A,B\}$ satisfies $2\le s\le n-1$ , where both inequalities are sharp, for every n. It is conjectured that the slot length of any irreducible pair of $n\times n$ matrices, where $n\ge 5$ , is at most $n-1$ . The slot length of a family of rank-one complex matrices can be equal to n.

中文翻译：

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矩阵族的时隙长度

我们引入的概念槽长任意域上的矩阵族${\mathbb {F}}$. 使用这个定义表明，如果$n\ge 5$和一种和乙是$n\次 n$复杂矩阵一种单细胞和对$\{A,B\}$不可约，槽长s的$\{A,B\}$满足$2\le s\le n-1$，其中两个不等式都很尖锐，对于每个n. 推测任何不可约对的槽长$n\次 n$矩阵，其中$n\ge 5$, 最多为$n-1$. 秩一复矩阵族的槽长度可以等于n.