  Quantum systems with a finite number of states at all times have been a
primary element of many physical models in nuclear and elementary particle
physics, as well as in condensed matter physics. Today, however, due to a
practical demand in the area of developing quantum technologies, a whole set of
novel tasks for improving our understanding of the structure of
finite-dimensional quantum systems has appeared. In the present article we will
concentrate on one aspect of such studies related to the problem of explicit
parameterization of state space of an $N$-level quantum system. More precisely,
we will discuss the problem of a practical description of the unitary
$SU(N)$-invariant counterpart of the $N$-level state space $\mathfrak{P}_N$,
i.e., the unitary orbit space $\mathfrak{P}_N/SU(N)$. It will be demonstrated
that the combination of well-known methods of the polynomial invariant theory
and convex geometry provides useful parameterization for the elements of
$\mathfrak{P}_N/SU(N)$. To illustrate the general situation, a detailed
description of $\mathfrak{P}_N/SU(N)$ for low-level systems: qubit $(N=2)\,,$
qutrit $(N=3)\,,$ quatrit $(N=4)\,$ - will be given.
