  We present the exact diagonalization study of rotating Bose-condensed gas
interacting via finite-range Gaussian potential confined in a quasi-2D harmonic
trap. The system of many-body Hamiltonian matrix is diagonalized in given
subspaces of quantized total angular momentum to obtain the lowest-energy
eigenstate employing the beyond lowest-Landau-level approximation. In the
co-rotating frame, the quantum mechanical stability of angular momentum states
is discussed for the existence of phase transition between the stable states of
interacting system. Thereby analyzing the von Neumann entanglement entropy and
degree of condensation provide the information about quantum phase correlation
in the many-body states. Calculating the conditional probability distribution,
we further probe the internal structure of quantum mechanically stable and
unstable states. Much emphasis is put on finding the spatial correlation of
bosonic atoms in the rotating system for the formation and entry of singly
quantized vortices, and then organizing into canonical polygons with and
without a central vortex at the trap center. Results are summarized in the form
of a movie depicting the vortex patterns having discrete p-fold rotational
symmetry with $p = 2,3,4,5,6$.
