The combinatorics of normal subgroups in the unipotent upper triangular group

Abstract

Uniformly describing the conjugacy classes of the unipotent upper triangular groups UT n (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 UT n (F q ). For q a prime, a bijection will be established between these subgroups and pairs of combinatorial objects with labels from F × q . 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 ut n (F q ) under an approximation of the exponential map.