  In nuclear theory, the generator coordinate method (GCM), a type of
configuration mixing method, is often used for the microscopic description of
collective motions. However, the GCM has a problem that a structure of the
collective subspace, which is the Hilbert space spanned by the configurations,
is not generally understood. In this paper, I investigate the structure of the
collective subspace in the dynamical GCM (DGCM), an improved version of the
GCM. I then show that it is restricted to a specific form that combines tensor
products and direct sums under reasonable conditions. By imposing additional
specific conditions that are feasible in actual numerical calculations, it is
possible to write the collective subspace as a simple tensor product of the
collective part and the others. These discussions are not dependent on the
details of the function space used for generating the configurations and can be
applied to various methods, including the mean-field theory. Moreover, this
analytical technique can also be applied to a variation after projection method
(VAP), then which reveals that under a specific condition, the function space
of the VAP has an untwisted structure. These consequences can provide powerful
tools for discussing the collective motions with the DGCM or the GCM.
