  The consequences of enforcing permutational symmetry, as required by the
Pauli principle (spin-statistical theorem), on the state space of molecular
ensembles interacting with the quantized radiation mode of a cavity are
discussed. The Pauli-allowed collective states are obtained by means of group
theory, i.e., by projecting the state space onto the appropriate irreducible
representations of the permutation group of the indistinguishable molecules. It
is shown that with increasing number of molecules the ratio of Pauli-allowed
collective states decreases very rapidly. Bosonic states are more abundant than
fermionic states, and the brightness of Pauli-allowed state space (the
contribution from photon excited states) increases(decreases) with increasing
fine structure in the energy levels of the material ground(excited) state
manifold. Numerical results are shown for the realistic example of rovibrating
H$_2$O molecules interacting with an infrared cavity mode.
