Exploring Group Theory in Theoretical Physics: From SU(2) to SU(5) Symmetries

This paper explores the implications of group theory in theoretical physics, focusing on the transition from SU(2) to SU(5) symmetries. We discuss the Weyl group's significance over $\mathbb{F}_1$ and its role in the "amplitudes=combinatorial geometry" program. We analyze the SU(5) group's properties and compare the SO(2) and SO(3) groups, explaining their evolution due to non-removable singularities in 3D spacetime. We examine the SO(2) symmetry of the Bumblebee Lagrangian and its Lorentz symmetry breaking implications. A proof for extending SU(2) to SU(5) via a reducible singularity expansion is presented. Finally, we discuss extending the Bumblebee Lagrangian to SU(5) symmetry. This paper highlights the importance of group theory in formulating complex physical models.This paper analyzes the conditions under which Hod's conjecture is violated in the context of Gauss-Bonnet-Bumblebee theory. We demonstrate that the Lagrangian satisfies the SO(2) symmetry group and that the violation of Hod's conjecture is due to conditions satisfying the SO(3)/SO(2) group. The analysis includes the impacts of Lorentz violation and Gauss-Bonnet coupling parameters on black hole quasinormal modes (QNMs) and the related optical properties.