论文标题
单位上三角组中正常亚组的组合学
The combinatorics of normal subgroups in the unipotent upper triangular group
论文作者
论文摘要
描述单位上三角组的共轭类$ \ mathrm {ut} _ {n}（\ mathbb {f} _ {q}）$均匀（对于$ n $和$ q $的所有或多个值）几乎是不可能的任务。本文提出了描述$ \ mathrm {ut} _ {n}（\ Mathbb {f} _ {q}）$的普通子组的相关问题。对于$ q $ a prime，将在这些子组和成对的组合对象之间建立两次两者，并带有来自$ \ mathbb {f} _ {q}^}^{\ times} $的标签。每对均包含一个无环的二进制矩阵和一个紧密的剪接，这是一种显然是新型的组合物体，可在非固定分区和缩短的多支着的多元中插入。对于任意$ Q $，相同的方法描述了正常子组的自然子集：与Lie代数$ \ Mathfrak {ut} _ {n}（\ Mathbb {f} _ {q}）$相对的lie代数$ \ mathfrak {ut} _ {n}）$的自然子集。
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.
