  We report on single-object spectroscopic studies of a characteristic set of
three resonances which appear in GaAlAs/AlAs quantum dot structures. The
experiments included the photoluminescence (PL) and photoluminescence
excitation (PLE) spectroscopy, photon-correlation measurements, analysis of the
linear polarization of the emission spectra as well as the PL and PLE
spectroscopy performed as a function of magnetic field. The investigated
resonances are assigned to three different states of a positively charged
exciton: a singlet ground state, a conventional triplet state (involving an
electron from the ground state electronic $s$-shell), and additionally, a
highly excited triplet state (involving the electron from the excited
electronic $p$-shell). Advantageously (due to partially suppressed relaxation
efficiency), the emission due to the highly excited triplet can seen in the
spectra which allows us to scrupulously determine its properties and compare
them to those of the conventional triplet state. Whereas both triplet states
show similar patterns of the fine structure and Zeeman splitting, the splitting
amplitudes are found to be significantly different for these two states. The
presented results emphasize the role of the symmetry of the electronic state on
the properties of the triplet states of the two holes + electron excitonic
complex.
