  The effective bosonic hamiltonian for excitons, extensively quoted up to now,
cannot be correct because it is (surprisingly) non-hermitian. The oversight
physically originates from the intrinsic difficulty of properly defining
electron-hole interactions between excitons when dealing with exchange terms.
By using our commutation technique, we show that the fermionic character of the
excitons cannot be forced into a dressed Coulomb interaction only : The
effective bosonic hamiltonian must contain purely fermionic terms of the same
order as the Coulomb terms. They are necessary to ensure hermiticity, and they
do not reduce to a two-body interaction, Pauli exclusion being N-body by
essence.
