Moriah and Schultens have
demonstrated that an irreducible Heegaard splitting of an orientable Seifert fibered
space over an orientable base surface is either vertical or horizontal. In this paper it is
determined precisely which vertical and horizontal splittings are irreducible. Let M
be a Seifert fibered space which admits a horizontal splitting at the fiber f. If the
genus of the horizontal splitting at f is less than the genus of the vertical splittings,
its genus will be minimal and the splitting irreducible. Otherwise, this splitting will
be irreducible if and only if the multiplicity of the fiber f is strictly greater
than the least common multiple of the multiplicities of the other fibers. In
particular, each Seifert fibered space possesses at most one irreducible horizontal
splitting. The vertical splittings will be reducible if and only if M has a
horizontal splitting with genus strictly less than the genus of the vertical
splittings.
