  We show that the knot group of any knot in any integer homology sphere admits
a non-abelian representation into $SU(3)$ such that meridians are mapped to
matrices whose eigenvalues are the three distinct third roots of unity. This
answers the $N=3$ case of a question posed by Xie and the first author. We also
characterize when a $PU(3)$-bundle admits a flat connection. The key ingredient
in the proofs is a study of the ring structure of $U(3)$ instanton Floer
homology of $S^1\times \Sigma_g$. In an earlier paper, Xie and the first author
stated the so-called eigenvalue conjecture about this ring, and in this paper
we partially resolve this conjecture. This allows us to establish a surface
decomposition theorem for $U(3)$ instanton Floer homology of sutured manifolds,
and then obtain the mentioned topological applications. Along the way, we prove
a structure theorem for $U(3)$ Donaldson invariants, which is the counterpart
of Kronheimer and Mrowka's structure theorem for $U(2)$ Donaldson invariants.
We also prove a non-vanishing theorem for the $U(3)$ Donaldson invariants of
symplectic manifolds.
