  For the first time, the calculation of the nuclear matrix element of the
double-$\beta$ decay of $^{100}$Mo, with and without the emission of two
neutrinos, is performed in the framework of the nuclear shell model. This task
is accomplished starting from a realistic nucleon-nucleon potential, then the
effective shell-model Hamiltonian and decay operators are derived within the
many-body perturbation theory. The exotic features which characterize the
structure of Mo isotopes -- such as shape coexistence and triaxiality softness
-- push the shell-model computational problem beyond its present limits, making
it necessary to truncate the model space. This has been done with the goal to
preserve as much as possible the role of the rejected degrees of freedom in an
effective approach that has been introduced and tested in previous studies.
This procedure is grounded on the analysis of the effective single-particle
energies of a large-scale shell-model Hamiltonian, that leads to a truncation
of the number of the orbitals belonging to the model space. Then, the original
Hamiltonian generates a new one by way of a unitary transformation onto the
reduced model space, to retain effectively the role of the excluded
single-particle orbitals. The predictivity of our calculation of the nuclear
matrix element for the neutrinoless double-$\beta$ decay of $^{100}$Mo is
supported by the comparison with experiment of the calculated spectra,
electromagnetic transition strengths, Gamow-Teller transition strengths and the
two-neutrino double-beta nuclear matrix elements.
