  Using exact diagonalization, we investigate the many-body ground state for
vortex patterns in a rotating Bose-condensed gas of $N$ spinless particles,
confined in a quasi-two-dimensional harmonic trap and interacting repulsively
via finite-range Gaussian potential. The $N$-body Hamiltonian matrix is
diagonalized in given subspaces of quantized total angular momentum $L_{z}$, to
obtain the lowest-energy eigenstate. Further, the internal structure of these
eigenstates is analyzed by calculating the corresponding conditional
probability distribution. Specifically, the quantum mechanically stable as well
as unstable states in a co-rotating frame are examined in the moderately
rotating regime corresponding to angular momenta $4N \le L_{z} < 5N$ for $N=16$
bosons. In response to externally impressed rotation, patterns of singly
quantized vortices are formed, shaping into canonical polygons with a central
vortex at the trap center. The internal structure of unstable states reveals
the mechanism of entry, nucleation and pattern formation of vortices with
structural phase transition, as the condensate goes from one stable vortical
state to the other. The stable polygonal vortex patterns having discrete
$p$-fold rotational symmetry with $p=5$ and $p=6$ are observed. The hexagonal
vortex pattern with $p=6$ symmetry is a precursor to the triangular vortex
lattice of singly quantized vortices in the thermodynamic limit. For unstable
states, quantum melting of vortex patterns due to uncertainty in positions of
individual vortices, is also briefly discussed.
