Basic characters of the unitriangular group (for arbitrary primes)

Let $U_{n}(q)$ denote the (upper) unitriangular group of degree $n$ over the finite field $\mathbb{F}_{q}$ with $q$ elements. In this paper we consider the basic (complex) characters of $U_{n}(q)$ and we prove that every irreducible (complex) character of $U_{n}(q)$ is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption $p \geq n$, where $p$ is the characteristic of the field $\mathbb{F}_{q}$.