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 $\mathrm{D}_6(q)$ and $\mathrm{E}_6(q)$, for an arbitrary power $q$ of any prime $p$. In particular, we establish that the parametrization is uniform for $p \ge 3$ in type $\mathrm{D}_6$ and for $p \ge 5$ in type $\mathrm{E}_6$, while the prime $2$ in type $\mathrm{D}_6$ and the primes $2,$ $3$ in type $\mathrm{E}_6$ 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 $\mathrm{D}_6$.