论文标题
线性关系与不相交的支持和内核的平均大小
Linear relations with disjoint supports and average sizes of kernels
论文作者
论文摘要
我们研究了矩阵模块中施加线性关系对核的平均大小的影响。我们认为的关系可以用网格的部分色素来形容。这些网格的细胞对应于矩阵中的位置，每个定义关系都涉及给定颜色的所有细胞。我们证明，与离散估值环有限的商相比，施加了“可接受”的部分色彩所产生的这种关系对内核的平均大小没有影响。这概括了以下事实，即普通广场的平均大小和相同尺寸的无纹子矩阵在此类环上重合。作为一个团体理论应用程序，我们明确确定Zeta函数列举了有限$ P $ - 组的共轭类别的类别，该类别从免费类别的$ 3 $ -NILPOTENT GROUPS衍生而来，用于$ p \ geqslant 5 $。
We study the effects of imposing linear relations within modules of matrices on average sizes of kernels. The relations that we consider can be described combinatorially in terms of partial colourings of grids. The cells of these grids correspond to positions in matrices and each defining relation involves all cells of a given colour. We prove that imposing such relations arising from "admissible" partial colourings has no effect on average sizes of kernels over finite quotients of discrete valuation rings. This vastly generalises the known fact that average sizes of kernels of general square and traceless matrices of the same size coincide over such rings. As a group-theoretic application, we explicitly determine zeta functions enumerating conjugacy classes of finite $p$-groups derived from free class-$3$-nilpotent groups for $p \geqslant 5$.
