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The combinatorics of normal subgroups in the unipotent upper triangular group

  • Author(s): Gagnon, Lucas
  • et al.

Published Web Location Commons 'BY' version 4.0 license

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

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