Tensor Product of Representations

This lecture explores the construction of matrices representing group representations, focusing on the existence of a base where the 6x6 matrices have a block structure. The instructor discusses the decomposition into irreducible representations and the calculation of characters to determine the structure of the matrices. The lecture delves into the concept of symmetry breaking and the transformation properties of different sectors under rotations, emphasizing the importance of group theory in simplifying the analysis of physical systems. The diagonalization of matrices and the selection rules for vibration modes are also covered, showcasing how symmetry considerations lead to distinct sectors of solutions. The lecture concludes with a demonstration of how different sectors transform uniquely under group operations, highlighting the fundamental role of symmetry in characterizing physical phenomena.