Combinatorial Theory

Uniformly describing the conjugacy classes of the unipotent upper triangular groups \(\mathrm{UT}_{n}(\mathbb{F}_{q})\) (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 set 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.

Keywords: Unipotent group, normal subgroup, Lie algebra ideal, nonnesting set partition, matroid, q-Stirling number.

Mathematics Subject Classifications: 05E16, 20G40, 17B45, 20E15