  We introduce certain $C^*$-algebras and $k$-graphs associated to $k$ finite
dimensional unitary representations $\rho_1,...,\rho_k$ of a compact group $G$.
We define a higher rank Doplicher-Roberts algebra
$\mathcal{O}_{\rho_1,...,\rho_k}$, constructed from intertwiners of tensor
powers of these representations. Under certain conditions, we show that this
$C^*$-algebra is isomorphic to a corner in the $C^*$-algebra of a row finite
rank $k$ graph $\Lambda$ with no sources. For $G$ finite and $\rho_i$ faithful
of dimension at least $2$, this graph is irreducible, it has vertices $\hat{G}$
and the edges are determined by $k$ commuting matrices obtained from the
character table of the group. We illustrate with some examples when
$\mathcal{O}_{\rho_1,...,\rho_k}$ is simple and purely infinite, and with some
$K$-theory computations.
