AI 中文总结
本研究计算有限矩阵群的 σ-矩生成函数，得出 Bₙ/Cₙ 型、Dₙ 型 Weyl 反射群的渐近 Λ-分布，建立 σ-矩生成函数形状与经典随机变量分布的关联结论。
AI 中文摘要
紧矩阵群的 Λ-分布是代数概率论中的一个不变量，近年被引入用于研究函数域 L-函数的零点分布，它由 σ-矩生成函数编码，是经典不变理论中 Molien 级数的推广。本工作计算了有限矩阵群在诸多新情形下的 σ-矩生成函数，尤其计算了 Bₙ/Cₙ 型和 Dₙ 型 Weyl 反射群无限族的渐近 Λ-分布，补充了此前已有的 Aₙ 型（即对称群）反射群的相关结果；还建立了将 σ-矩生成函数的形状与相关经典随机变量分布关联起来的一般性结论，解释了此前关于紧经典群上幂迹产生的独立高斯变量的特设观测。
英文摘要
The $Λ$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $σ$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $σ$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $Λ$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $σ$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.
Comments18 pages, comments welcome. v2 corrects Theorem B, part (2)
