On the Character Degrees of Sylow $p$-subgroups of Chevalley Group of Type $E(p^f)$

TL;DR

This paper constructs specific irreducible characters of Sylow p-subgroups in Chevalley groups of types D4, E6, and E8, revealing how certain primes are 'bad' primes affecting their representation theory.

Contribution

It provides explicit constructions of irreducible characters with unusual degrees for Sylow p-subgroups of exceptional Chevalley groups, clarifying the role of bad primes in their representation theory.

Findings

Constructed characters of degree q^3/2 for D4(q) with q=2^f.

Constructed characters of degree q^7/3 for E6(q) with q=3^f.

Constructed characters of degree q^{16}/5 for E8(q) with q=5^f.

Abstract

Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of the irreducible characters of the Sylow $p$-subgroup of $GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed that this is true for a Sylow $p$-subgroup of a classical group defined over $\F_q$ if and only if $p$ is odd. For the classical groups of Lie type $B$, $C$ and $D$ the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow $p$-subgroups of the Chevalley groups $D_4(q)$ with $q=2^f$ of degree $q^3/2$. Then we use an analogous construction for $E_6(q)$ with $q=3^f$ to obtain characters of degree $q^7/3$, and for $E_8(q)$ with $q=5^f$ to obtain characters of degree $q^{16}/5.$ This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type $E$ in terms of the representation theory of the Sylow $p$-subgroup.