  It is shown that the layered antiferromagnetic order in stoechiometric
LaMnO_3 cannot be understood purely from electronic interactions. On the
contrary, it mainly results from strong cooperative Jahn-Teller deformations.
Those involve a compression of the Mn-O octahedron along the c-axis (mode Q_3 <
0), while alternate Jahn-Teller deformations occur in the ab-plane (mode Q_2).
These deformations stabilize a certain type of orbital ordering. The resulting
superexchange couplings are calculated by exact diagonalization, taking into
account both e_g and t_{2g} orbitals. The main result is that antiferromagnetic
(ferromagnetic) coupling along the c-direction (ab-planes) can be understood
only if the Jahn-Teller energy is much larger than the superexchange couplings,
which is consistent with experiments. This mechanism contrasts with that based
on weak Jahn-Teller coupling which instead predicts elongation along the c-axis
(Q_3 > 0). The crucial role of the deformation anisotropy Q_2/Q_3 is also
emphasized.
