Given a finite group $G$ with irreducible character
$\chi \in \Irr(G), \hspace{1ex} r \in {\mathbb N}$
and a partition $\lambda$ % \vdash of $r$,
we define higher characters $\chi_{\lambda}^{(r)}$ of $G$,
following Frobenius \cite{Frob}.
We interpret them as a generalization of Schur functions
in noncommuting variables, as a multilinear invariant map,
as the sum over $\S_r$ of a character of the wreath product
$G \wr \S_r$, and as the trace of
a $e_\lambda G_r e_\lambda$-module.
Using these interpretations, we are able to compute
$(\chi + \psi)_\lambda^{(r)}$ in terms of $\chi_\mu^{(a)}$ and
$\psi_\nu^{(b)}$ where $a+b=r$ and $\mu$ and $\nu$ are partitions
of $a$ and $b$, respectively.
We show distinct higher characters are orthogonal
in section \ref{orthog_sec}.
These $\chi_{\lambda}^{(r)}$ have the property that
they are constanton $\S_r \times G$ orbits of $G^r$,
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i.e.
invariant under diagonal conjugation and permutation of entries
of an r-tuple.
By decomposing ${\Hom}_{G_r}({\Ind}_{\S_r}^{G_r} E_{\lambda},
{\Ind}_{\S_r}^{G_r} E_{\lambda}) = e_\lambda G_r e_\lambda$,
we find an orthogonal family of functions with this property.
In the case $G$ is abelian, we show this family forms a basis
for all such functions on $G^r$.
This doesn't happen for general $G$, as shown on some
examples in the appendix.
However, in both cases, we show that it is only necessary
to consider $\lambda =
(r)$ (or $\lambda = (1^r)$)
in theorems \ref{bas} and \ref{abbas}.
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The reader may skip
straight to section \ref{subsec_w} for these results.