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

TL;DR

This paper parametrizes irreducible characters of Sylow p-subgroups of Chevalley groups D6 and E6 over finite fields, revealing uniformity for large primes and new character degree phenomena for small primes.

Contribution

It provides a uniform parametrization of these characters for large primes and uncovers novel character degree structures for small primes in types D6 and E6.

Findings

Parametrization is uniform for p ≥ 3 in D6 and p ≥ 5 in E6.

Character degrees include new forms like q^m/p^i with i > 1.

Small primes cause deviations from generic character degree patterns.

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 $\mathrm{D}_6$.