  We show that the hypothesis of abelian dominance in maximal abelian gauge,
which is known to work for Wilson loops in the fundamental representation,
fails for Wilson loops in higher group representations. Monte Carlo simulations
are performed on lattice $SU(2)$ gauge theory, in $D=3$ dimensions, in the
maximal abelian gauge, in the confined phase. It is well-known that Creutz
ratios extracted from loops in various group representations are proportional
to the quadratic Casimir of each representation, in a distance interval from
the confinement scale to the point where color screening sets in. In contrast
we find numerically, in the same interval, that string tensions extracted from
loops built from abelian projected configurations are the same for the
fundamental and $j=3/2$ representations, and vanish for the adjoint
representation. In addition, we perform a lattice Monte Carlo simulation of the
Georgi-Glashow model in $D=3$ dimensions. We find that the
representation-dependence of string tensions is that of pure Yang-Mills in the
symmetric phase, but changes abruptly to equal tensions for the $j=1/2,~3/2$
representations, and zero tension for $j=1$, at the transition to the Higgs
phase. Our results indicate that an effective abelian theory at the confinement
scale, invoking {\it only} degrees of freedom (monopoles and photons)
associated with a particular Cartan subalgebra, is inadequate to describe the
actual interquark potential in an unbroken non-abelian gauge theory.
