Constructing characters of Sylow p-subgroups of finite Chevalley groups

Abstract

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...

On the characters of Sylow $p$-subgroups of finite Chevalley groups $G(p^f)$ for arbitrary primes

“…In this work, we first develop a parametrization of Irr(U ) by means of positive root sets of G, which is valid for arbitrary primes. This procedure generalizes the one in [7] and [18], which does not work for type F 4 when p = 2. In general, if p is a very bad prime for G then we lose some structural information when passing from patterns to pattern groups.…”

“…For type A 4 there are no nonabelian cores at any prime. As in [7,Section 4], there is just one [3,10,9]-core in type B 4 for p ≥ 3 and in type D 4 for arbitrary primes, there are 6 nonabelian cores of different forms in type F 4 for p ≥ 3, and there are no nonabelian cores in type C 4 for p ≥ 3. In the case of UB 4 (2 f ) ∼ = UC 4 (2 f ) we have 51 nonabelian cores of the form [2,4,1], one [4,8,2]-core and one [4,11,6]-core.…”

“…If N 1 = N 2 and P 1 ⊆ P 2 , for T := P 2 \ P 1 we define Ind T to be the induction from X S 1 to X S 2 , and if T = {α} we put Ind α := Ind T . 7 We need the following adaptation of [7, Lemma 3.1]. The proof repeats mutatis mutandis.…”

“…We now proceed to illustrate the adaptation of [7,Algorithm 3.3]. At each step, we assume that the tuple C = (S, Z, A, L, K) is constructed and currently taken into consideration.…”

“…In Section 3, we give the definition of pattern and quattern groups valid for all primes. In Section 4, we generalize [7,Algorithm 3.3] to obtain all abelian and nonabelian cores. We discuss in Section 5 on the…”