Group Theoretical Analysis in Modern Physics: Extending Symmetries from SU(2) to SU(5) and Implications in Theoretical Models

The physical significance of the Weyl group, a reductive group over $\mathbb{F}_1$, is highlighted through several pieces of evidence. Additionally, the work initiated by Arkani-Hamed et al. in 2013, known as the ``amplitudes=combinatorial geometry" program, demonstrates a combinatorial character in calculating amplitudes. This approach notably simplifies the complex and exponentially growing calculations of Feynman diagrams within the $\mathcal{S O}(2)$ framework\cite{1,2}. This paper delves into the intricate application of group theory in theoretical physics, emphasizing the transition of symmetries from SU(2) to SU(5). We begin by examining the foundational properties of the SU(5) group, including aspects such as closure, associativity, identity, inverse elements, and unitarity. This exploration lays the necessary groundwork for applying these principles in advanced theoretical physics.A comparative study of the SO(2) and SO(3) groups is then presented, highlighting the impact of non-removable singularities in 3D spacetime on these groups. This comparison is crucial for understanding the necessity of transitioning from SO(2) to SO(3) or other higher-dimensional SO groups, thereby elucidating the role of group theory in spatial rotations and symmetries within physics.The extension of the Bumblebee Lagrangian from SO(2) to SU(5) symmetry is explored in the final section. This process involves the introduction of additional fields and symmetries to integrate the higher-dimensional structure of the SU(5) group, enhancing the existing model with increased complexity and potential for novel physical interpretations.