有限域上矩阵的Jordan-Landau定理
Jordan--Landau theorem for matrices over finite fields
摘要：
给定一个正整数$r$和一个素数幂$q$，当$n$较大时，我们估计了随机矩阵$A$在$\mathrm{GL}_{n}(\mathbb{F}_{q})$中的特征多项式$f_{A}(t)$是无平方因子且有$r$（monic）不可约因子的概率.我们还估计了类似的概率，$f_{A}(t)$有$r$不可约因子计数的多重性。在任何一种情况下，主项$(\log n)^{r-1}((r-1)!n)^{-1}$和误差项$O((\log n)^{r-2}n^{-1})$（其隐含常数仅取决于$r$，而不取决于$q$或$n$）与$n$字母上的随机置换是$r$不相交循环的乘积的概率一致。我们证明的主要成分是一个递归参数由于S。D. Cohen，它以前用于估计$\mathbb{F}_{q}[t]$中的随机度$n$一元多项式是无平方的且具有$r$不可约因子的概率，以及多项式具有$r$不可约因子且具有多重性的类似概率。我们得到我们的结果通过仔细修改科恩的递归参数在矩阵设置，使用赖纳定理，计数的数目$n \times n$矩阵与一个固定的特征多项式超过$\mathbb{F}_{q}$。
英文摘要：
Given a positive integer $r$ and a prime power $q$, we estimate the probability that the characteristic polynomial $f_{A}(t)$ of a random matrix $A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic) irreducible factors when $n$ is large. We also estimate the analogous probability that $f_{A}(t)$ has $r$ irreducible factors counting with multiplicity. In either case, the main term $(\log n)^{r-1}((r-1)!n)^{-1}$ and the error term $O((\log n)^{r-2}n^{-1})$, whose implied constant only depends on $r$ but not on $q$ nor $n$, coincide with the probability that a random permutation on $n$ letters is a product of $r$ disjoint cycles. The main ingredient of our proof is a recursion argument due to S. D. Cohen, which was previously used to estimate the probability that a random degree $n$ monic polynomial in $\mathbb{F}_{q}[t]$ is square-free with $r$ irreducible factors and the analogous probability that the polynomial has $r$ irreducible factors counting with multiplicity. We obtain our result by carefully modifying Cohen's recursion argument in the matrix setting, using Reiner's theorem that counts the number of $n \times n$ matrices with a fixed characteristic polynomial over $\mathbb{F}_{q}$.
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