Involutions and characters of upper triangular matrix groups

We study the realizability over $\mathbb{R}$ of representations of the group $U(n)$ of upper-triangular $n \times n$ matrices over $\mathbb{F} _2$. We prove that all the representations of $U(n)$ are realizable over $\mathbb R$ if $n \leq 12$, but that if $n \geq 13$, $U(n)$ has representations not realizable over $\mathbb R$. This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-López and of the authors, but the proof of the theorem in this paper is much more natural.