  Let $\UT_n(q)$ denote the unitriangular group of unipotent $n\times n$ upper
triangular matrices over a finite field with cardinality $q$ and prime
characteristic $p$. It has been known for some time that when $p$ is fixed and
$n$ is sufficiently large, $\UT_n(q)$ has ``exotic'' irreducible characters
taking values outside the cyclotomic field $\QQ(\zeta_p)$. However, all proofs
of this fact to date have been both non-constructive and computer dependent. In
a preliminary work, we defined a family of orthogonal characters decomposing
the supercharacters of an arbitrary algebra group. By applying this
construction to the unitriangular group, we are able to derive by hand an
explicit description of a family of characters of $\UT_n(q)$ taking values in
arbitrarily large cyclotomic fields. In particular, we prove that if $r$ is a
positive integer power of $p$ and $n>6r$, then $\UT_n(q)$ has an irreducible
character of degree $q^{5r^2-2r}$ which takes values outside $\QQ(\zeta_{pr})$.
By the same techniques, we are also able to construct explicit Kirillov
functions which fail to be characters of $\UT_n(q)$ when $n>12$ and $q$ is
arbitrary.
