# Polynomial properties of unipotent radicals of parabolic subgroups in classical groups

## TL;DR

This paper derives explicit formulas for counting irreducible characters of unipotent radicals in classical groups, enhancing understanding of their representation theory.

## Contribution

It provides new explicit formulas for the number of irreducible characters of unipotent radicals in classical groups, a previously unquantified aspect.

## Key findings

- Formulas for irreducible character counts in symplectic groups
- Formulas for irreducible character counts in orthogonal groups
- Formulas for irreducible character counts in unitary groups

## Abstract

Let $R_u^{X,d}$ denote the unipotent radical of a (proper) maximal standard parabolic subgroup of the classical group $\mathrm{Sp}_{2n}(q)$, $\mathrm{SO}_{2n}(q)$, or $\mathrm U_{2n}({q^2})$. This paper establishes explicit formulas for the number of irreducible characters of $R_u^{X,d}$ with degree $q^e$.