Constructing characters of Sylow p-subgroups of finite Chevalley groups

Let q be a power of a prime p, let G   be a finite Chevalley group over FqFq 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 F4F4, where we observe that the parametrization is “uniform” over good primes p>3p>3, but differs for the bad prime p=3p=3. We also explain how it has been applied for all groups of rank 4 or less.