Characters of the Sylow p-Subgroups of the Chevalley Groups D_4(p^n)

TL;DR

This paper classifies all complex irreducible characters of Sylow p-subgroups of Chevalley groups D_4(q), revealing their structure and degree multiplicities as polynomials in q-1 with nonnegative coefficients.

Contribution

It provides a complete construction and classification of irreducible characters of these Sylow p-subgroups, linking their degrees to polynomials in q-1.

Findings

All irreducible characters are constructed explicitly.

Degrees of characters are given by polynomials in q-1.

Multiplicities of degrees have nonnegative polynomial coefficients.

Abstract

Let $U(q)$ be a Sylow $p$-subgroup of the Chevalley groups $D_4(q)$ where $q$ is a power of a prime $p$. We describe a construction of all complex irreducible characters of $U(q)$ and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in $q-1$ with nonnegative coefficients.