  Let $UY_n(q)$ be a Sylow p-subgroup of an untwisted Chevalley group $Y_n(q)$
of rank n defined over $\mathbb{F}_q$ where q is a power of a prime p. We
partition the set $Irr(UY_n(q))$ of irreducible characters of $UY_n(q)$ into
families indexed by antichains of positive roots of the root system of type
$Y_n$. We focus our attention on the families of characters of $UY_n(q)$ which
are indexed by antichains of length 1. Then for each positive root $\alpha$ we
establish a one to one correspondence between the minimal degree members of the
family indexed by $\alpha$ and the linear characters of a certain subquotient
$\overline{T}_\alpha$ of $UY_n(q)$. For $Y_n = A_n$ our single root character
construction recovers amongst other things the elementary supercharacters of
these groups. Most importantly though this paper lays the groundwork for our
classification of the elements of $Irr(UE_i(q))$, $6 \le i \le 8$ and
$Irr(UF_4(q))$.
