The combinatorics of normal subgroups in the unipotent upper triangular group

TL;DR

This paper develops a combinatorial framework to describe normal subgroups of unipotent upper triangular groups over finite fields, linking algebraic structures to new combinatorial objects like matroids and splices.

Contribution

It introduces a novel combinatorial approach to classify normal subgroups of $ ext{UT}_n( extbf{F}_q)$, including a bijection with pairs of combinatorial objects for prime q.

Findings

Bijection between normal subgroups and combinatorial pairs for prime q

Introduction of 'tight splice' as a new combinatorial object

Description of normal subgroups related to Lie algebra ideals

Abstract

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.