  The neutron-rich $^{122-131}$Sb isotopes were produced as fission fragments
in the reaction $^{9}$Be($^{238}$U,~f) with 6.2 MeV/u beam energy. An unique
setup, consisting of AGATA, VAMOS++ and EXOGAM detectors, was used which
enabled the prompt-delayed gamma-ray ($\gamma$) spectroscopy of fission
fragments in the time range of 100 ns - 200 $\mu$s. New isomers, prompt and
delayed transitions were established in the even-A $^{122-130}$Sb isotopes. In
the odd-A $^{123-131}$Sb isotopes, new prompt and delayed $\gamma$-ray
transitions were identified, in addition to the confirmation of the previously
known isomers. The half-lives of the isomeric states and the $B(E2)$ transition
probabilities of the observed transitions depopulating these isomers were
extracted. The experimental data was compared with the theoretical results
obtained in the framework of Large-Scale Shell-Model (LSSM) calculations in a
restricted model space. Modifications of several components of the shell model
interaction were introduced to obtain a consistent agreement with the
excitation energies and the $B(E2)$ transition probabilities in neutron-rich Sn
and Sb isotopes. The isomeric configurations in Sn and Sb were found to be
relatively pure. Further, the calculations revealed that the presence of a
single valence proton, mainly in the $g_{7/2}$ orbital in Sb isotopes, leads to
significant mixing (due to the $\nu\pi$ interaction) of: (i) the neutron
seniorities ($\upsilon_{\nu}$) and (ii) the neutron angular momentum
($I_{\nu}$). The above features have a weak impact on the excitation energies,
but have an important impact on the $B(E2)$ transition probabilities. In
addition, a constancy of the relative excitation energies irrespective of
neutron seniority and neutron number in Sn and Sb was observed.
