The concept of invariance of social choice functions to the removal or addition of minimal maximally symmetric sub profiles is found in the works of Donald Saari. In the theory of social choice over strict linear profiles, Saari defines the concept of a kernel (vector) — a profile that contains one of each preference (permutation). Such a profile is maximally symmetric (each candidate appears an equal number of times in each position in the profile), but it is not minimal. There are profiles in which each candidate appears exactly once in each of the positions, which then makes such a profile a minimally symmetric profile. Saari calls such profiles Condorcet profiles, since they generate a non-symmetric majority vote cycle, and thus affect the result of the pairwise comparison method.While Saari’s framework has been extensively studied from an algebraic and geometric perspective, its implications for visualization and structural reduction of social choice domains have remained largely unexplored. In this paper, we show how the axiom of invariance on the addition of minimal maximally symmetric sub profiles can be used in the field of visualizing and simplifying the domains of social choice functions. Already in the situation with three candidates, we define social choice functions over a discrete six-dimensional profile domain, which is a structure that cannot be directly visualized in the form of a (pseudo) graph. We show that for social choice functions that satisfy the aforementioned axiom of invariance, the domain can be reduced to (at most) three dimensions, which allows for a pseudo-graphical representation of the results of the social choice function, the image of which is finite (which is the case for social choice functions, as well as for social welfare functions). This approach provides a novel way to connect algebraic invariance with geometric visualization, offering new tools for understanding the structure and behavior of social choice mechanisms.