The irreducible characters of the Sylow $p$-subgroups of the Chevalley groups $\mathrm{D}_6(p^f)$ and $\mathrm{E}_6(p^f)$

Abstract

We parametrize the set of irreducible characters of the Sylow p-subgroups of the Chevalley groups D 6 (q) and E 6 (q), for an arbitrary power q of any prime p. In particular, we establish that the parametrization is uniform for p ≥ 3 in type D 6 and for p ≥ 5 in type E 6 , while the prime 2 in type D 6 and the primes 2, 3 in type E 6 yield character degrees of the form q m /p i which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters of a classical group of degree q m /p i where i > 1 which occurs in type D 6 .