  We study linear free divisors, that is, free divisors arising as
discriminants in prehomogeneous vector spaces, and in particular in quiver
representation spaces. We give a characterization of the prehomogeneous vector
spaces containing such linear free divisors. For reductive linear free
divisors, we prove that the numbers of geometric and representation theoretic
irreducible components coincide. As a consequence, we find that a quiver can
only give rise to a linear free divisor if it has no (oriented or unoriented)
cycles. We also deduce that the linear free divisors which appear in Sato and
Kimura's list of irreducible prehomogeneous vector spaces are the only
irreducible reductive linear free divisors. Furthermore, we show that all
quiver linear free divisors are strongly Euler homogeneous, that they are
locally weakly quasihomogeneous at points whose corresponding representation is
not regular, and that all tame quiver linear free divisors are locally weakly
quasihomogeneous. In particular, the latter satisfy the logarithmic comparison
theorem.
