  We present a self-consistent theoretical framework for finite-dimensional
discrete phase spaces that leads us to establish a well-grounded mapping scheme
between Schwinger unitary operators and generators of the special unitary group
$\mathrm{SU(N)}$. This general mathematical construction provides a sound
pathway to the formulation of a genuinely discrete Wigner function for
arbitrary quantum systems described by finite-dimensional state vector spaces.
To illustrate our results, we obtain a general discrete Wigner function for the
group $\mathrm{SU(3)}$ and apply this to the study of a particular three-level
system. Moreover, we also discuss possible extensions to the discrete Husimi
and Glauber-Sudarshan functions, as well as future investigations on
multipartite quantum states.
