  We present magneto-optical studies of a self-assembled semiconductor quantum
dot, concentrating specifically on the case in which the dot is doubly
positively charged, studying this way the confined hole - hole exchange
interaction. A simple harmonic potential model, which we extend to capture the
influence of an externally applied magnetic field in Faraday configuration
fully describe the observed polarization sensitive magneto-photoluminscence
spectra. We deduce the effective composition of the quantum dot from its
measured electronic g-factor. Using this value we determine the dot effective
permittivity and quantitatively describe various measured excitonic
transitions, their measured Zeeman splittings and their diamagnetic shifts. In
particular, the model quantitatively accounts for an observed pronounced
negative diamagnetic shift, which provides a direct measure for the hole-hole
exchange interaction and its dependence on the externally applied magnetic
field strength.
