  It has been observed that the composite fermion (CF) approach tends to
overcount the number of linearly independent candidate states for fixed sets of
quantum numbers [number of particles, total angular momentum, and (pseudo)spin
if applicable]. That is, CF Slater determinants that are orthogonal before
projection, may lead to wave functions that are identical, or possess linear
dependencies, after projection. This has been pointed out both in the context
of rotating bosons in the lowest Landau level, and for excited bands of the
(fermionic) fractional quantum Hall effect. We present a systematic approach
that enables us to reveal all linear dependencies between bosonic compact
states in the lowest CF "cyclotron energy" sub-band, and almost all
dependencies in higher sub-bands, at the level of the CF Slater determinants,
i.e. before projection, which implies a major computational simplification. Our
approach is introduced for so-called simple states of two-species rotating
bosons, and then generalised to generic compact bosonic states, both one- and
two-species. Some perspectives also apply to fermionic systems. The identities
and linear dependencies we find, are analytically exact for "brute force"
projection in the disk geometry.
