论文标题
约旦类型的三角形矩阵在有限场上
Jordan types of triangular matrices over a finite field
论文作者
论文摘要
令$λ$为整数$ n $，$ {\ mathbb f} _q $的分区为订单$ q $的有限字段。令$p_λ（q）$为约旦型$λ$的严格上部三角形$ n \ times n $矩阵的数量。众所周知，多项式$p_λ$具有$ q $和$ q = q = q-1 $的高功率的趋势，我们将$p_λ（q）= q^{d（λ）} q^{e（λ）}r_λ（q）r_λ（q）$，其中$r_λ（q）$r_λ（0）\ neq0 $和$ r_ / q r_ r_ / q.0（1）。在本文中，我们研究了多项式$p_λ（q）$和$r_λ（q）$。我们的主要结果：$ d（λ）$的明确公式（已知$ e（λ）$的明确公式，请参见下面的命题3.3），$r_λ（q）$的递归公式（$r_λ（q）的类似公式（$p_λ（q）$的类似公式已知，请参阅下面的命题3.1，估计$r_λ$ a $r_λ$ a $r_λ$ viste $r_λ$λ限制系列的公式$ r _ {\ lambda1^\ infty} $。我们的研究是由对轨道法代表理论的轨道法对有限域的代表理论进行的预计应用的动机。
Let $λ$ be a partition of an integer $n$ and ${\mathbb F}_q$ be a finite field of order $q$. Let $P_λ(q)$ be the number of strictly upper triangular $n\times n$ matrices of the Jordan type $λ$. It is known that the polynomial $P_λ$ has a tendency to be divisible by high powers of $q$ and $Q=q-1$, and we put $P_λ(q)=q^{d(λ)}Q^{e(λ)}R_λ(q)$, where $R_λ(0)\neq0$ and $R_λ(1)\neq0$. In this article, we study the polynomials $P_λ(q)$ and $R_λ(q)$. Our main results: an explicit formula for $d(λ)$ (an explicit formula for $e(λ)$ is known, see Proposition 3.3 below), a recursive formula for $R_λ(q)$ (a similar formula for $P_λ(q)$ is known, see Proposition 3.1 below), the stabilization of $R_λ$ with respect to extending $λ$ by adding strings of 1's, and an explicit formula for the limit series $R_{\lambda1^\infty}$. Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.
