23 pagesInternational audienceWe identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras

Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is "uniform" over good primes p > 3, but differs for the bad prime p = 3. We also explain how it has been applied for all groups of rank 4 or less. 1In fact for G of rank 4 or less, we obtain that each irreducible character of U can be obtained by inducing a linear character. In addition, we remark that our labelling of the irreducible characters is amenable to the action of a maximal torus and also the field automorphisms; thus it would be straightforward to determine these actions explicitly.The methods in this paper develop those used by Himstedt and the second and third authors in [HLM1] and [HLM2], and make significant further progress. A full parametrization of the irreducible characters of U for G of type D 4 and every prime p is given in [HLM1]. The so called single root minimal degree almost faithful irreducible characters are parameterized for every type and rank when p is not a very bad prime for G in [HLM2].The approach used in those papers and this paper is built on partitioning the irreducible characters of U in terms of the root subgroups that lie in their centre, but not in their kernel. Consequently, there are similarities to the theory of supercharacters, which were first studied for the case G is of type A by André, see for example [An]. This theory was fully developed by Diaconis and Isaacs in [DI]. Subsequently it was applied to the characters of U for G of types B, C and D by André and Neto in [AN].Another approach to the character theory of U is via the Kirillov orbit method, which is applicable for p greater than the Coxeter number of G. In [GMR2], Mosch, Röhrle and the first author explain an algorithm for parameterizing the coadjoint orbits of U, which was applied for G of rank at most 8, except E 8 ; through the Kirillov orbit method this leads to a parametrization of the irreducible characters of U. This was preceded by an algorithm to determine the conjugacy classes of U, see [GMR1].We note that a reduction procedure for algebra groups similar to ours was given by Evseev in [Ev] and builds on work of Isaacs in [Is2]. For G = SL n (q), this led to a parametrization of the irreducible characters of U for n ≤ 13. Also recently Pak and Soffer have determined the coadjoint orbits of U for G = SL n (q) and n ≤ 16, see [PS]. The situation for G not of type A turns out to be more complicated and we comment more on this below.There has been considerable other interest in the character theory and conjugacy classes of U. We refer the reader to [LM1] or the introductio...

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In [J. Algebra 344 (2011), 205–228], a conjecture of J. G. Thompson for PSL

Let U n (q) be the upper triangular group of degree n over the finite field F q with q elements. In this paper, we present constructions of large degree (complex) irreducible representations of U n (q) where n 7, and then determine the number of irreducible representations of largest, second largest and third largest degrees.