Symmetries of conformal correlation functions

A program of wide interest in modern conformal bootstrap studies is to numerically solve general field theories, based on a critical assumption that the dynamics encoded four-point crossing equations and positivity condition. In this paper we propose verify novel algebraic property which provides strong restriction for program. We show various types symmetries $\mathcal{G}$, can be linearly converted into $SO(N)$ vector associated with $SO(N)\ensuremath{\rightarrow}\mathcal{G}$ branching rules transformations satisfy The constrained by $\mathcal{G}$-symmetric combined condition degenerates symmetric cases, while non-$SO(N)$ theories are not directly solvable without introducing symmetry breaking assumptions spectrum.