  In this paper, with the help of the $S$ function and ghost symmetry for the
discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the
ghost flow on its eigenfunction(adjoint eigenfunction) and the spectral
representation of its Baker-Akhiezer function and adjoint Baker-Akhiezer
function are derived. From these observations above, some important
distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also
we give the ghost flow on the tau function and another kind of proof of the
ASvM formula of the discrete KP hierarchy.
