We study equilibrium states of the Smoluchowski equation for rigid, rod-like polymer
ensembles. We start with several cases in the three dimensional space: a) nematic polymers where
the only intermolecular interaction is the excluded volume effect, modeled using the Maier-Saupe
potential, b) dipolar nematic polymers where the intermolecular interaction consists of the dipole-
dipole potential and the Maier-Saupe potential, c) dipolar nematic polymers in the presence of a
stretching elongational flow, and d) nematic polymers in higher dimensional space. For each of the
cases a), b) and c), it has been established separately with various mathematical manipulations
that all stable equilibrium states have rotational symmetry. In this study, we present a unified
view of the rotational symmetry of cases a), b) and c). Specifically, in cases a), b) and c), the
rotational symmetry is determined by a key inequality. The inequality, once established for case a),
is extended elegantly to cases b) and c). Furthermore, this inequality is used in case d) to establish
the rotational symmetry of equilibrium states of nematic polymers in higher dimensional space. In
three dimensional space, rotational symmetry simply means axisymmetry. In higher dimensional
space, rotational symmetry is more complex in structure. For example, in four dimensional space,
rotational symmetry may be around a one dimensional sub-space (i.e., axisymmetry) or it may be
around a two dimensional sub-space. Nevertheless, the rotational symmetry significantly simplifies
the classification of equilibrium states. We calculate and present phase diagrams of nematic polymers
in higher dimensional spaces.