{"pk":64811,"title":"The combinatorics of normal subgroups in the unipotent upper triangular group","subtitle":null,"abstract":"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.\nKeywords: Unipotent group, normal subgroup, Lie algebra ideal, nonnesting set partition, matroid, q-Stirling number.\nMathematics Subject Classifications: 05E16, 20G40, 17B45, 20E15","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/5c21s1xq","frozenauthors":[{"first_name":"Lucas","middle_name":"","last_name":"Gagnon","name_suffix":"","institution":"Department of Mathematics, University of Colorado Boulder, Colorado, U.S.A.","department":""}],"date_submitted":"2021-11-12T21:44:58Z","date_accepted":"2021-11-12T21:44:58Z","date_published":"2021-12-15T08:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64811/galley/49621/download/"}]}