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 ${\mathrm{D}}_6$.

32 pages