标题： 单位上三角群中正规子群的组合学 显示英文标题

标题： The combinatorics of normal subgroups in the unipotent upper triangular group

摘要： 描述不可约上三角群的共轭类$\mathrm{UT}_{n}(\mathbb{F}_{q})$在所有或许多值的$n$和$q$下统一地进行几乎是不可能的任务。本文解决了描述$\mathrm{UT}_{n}(\mathbb{F}_{q})$的正规子群的相关问题。对于$q$为素数的情况，将建立这些子群与来自$\mathbb{F}_{q}^{\times}$的标签的组合对象对之间的双射。每对包括一个无环二元拟阵和一个紧密拼接，这是一种似乎全新的组合对象，它在非嵌套划分和缩短多米诺骨牌之间进行插值。 对于任意的$q$，相同的方法描述了正规子群的一个自然子集：这些正规子群对应于李代数$\mathfrak{ut}_{n}(\mathbb{F}_{q})$的理想，在指数映射的近似下。 显示英文摘要

摘要： Describing the conjugacy classes of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ uniformly (for all or many values of $n$ and $q$) is a nearly impossible task. This paper takes on the related problem of describing the normal subgroups of $\mathrm{UT}_{n}(\mathbb{F}_{q})$. For $q$ a prime, a bijection will be established between these subgroups and pairs of combinatorial objects with labels from $\mathbb{F}_{q}^{\times}$. Each pair comprises a loopless binary matroid and a tight splice, an apparently new kind of combinatorial object which interpolates between nonnesting partitions and shortened polyominoes. For arbitrary $q$, the same approach describes a natural subset of normal subgroups: those which correspond to the ideals of the Lie algebra $\mathfrak{ut}_{n}(\mathbb{F}_{q})$ under an approximation of the exponential map.