  We here consider an exciton $i$ embedded in a sea of $N$ identical excitons
0. If the excitons are bosonized, a bosonic enhancement factor, proportional to
$N$, is found for $i=0$. If the exciton composite nature is kept, this
enhancement not only exists for $i=0$, but also for any exciton having a center
of mass momentum equal to the sea exciton momentum. This physically comes from
the fact that an exciton with such a momentum can be transformed into a sea
exciton by ``Pauli scattering'', \emph{i}. \emph{e}., carrier exchange with the
sea, making this $i$ exciton not so much different from a 0 exciton. This
possible scattering, directly linked to the composite nature of the excitons,
is irretrievably lost when the excitons are bosonized.
  This work in fact deals with the quite tricky scalar products of $N$-exciton
states. It actually constitutes a crucial piece of our new many-body theory for
interacting composite bosons, because all physical effects involving these
composite bosons ultimately end by calculating such scalar products. The
``Pauli diagrams'' we here introduce to represent them, allow to visualize
many-body effects linked to carrier exchange in an easy way. They are
conceptually different from Feynman diagrams, because of the special feature of
the ``Pauli scatterings'': These scatterings, which originate from the
departure from boson statistics, do not have their equivalent in Feynman
diagrams, the commutation rules for exact bosons (or fermions) being included
in the first line of the usual many-body theories.
