Unitary representations

A unitary representation is a way to describe symmetry in mathematical terms using matrices that preserve lengths and angles. Imagine a shape that can be rotated or reflected without changing its size or form; these transformations can be represented mathematically as matrices. When these matrices are "unitary," they maintain the overall structure—meaning they don't distort the shape's size or angles—making them essential for analyzing symmetrical structures in quantum mechanics, signal processing, and other fields. Essentially, unitary representations help us study complex symmetries while preserving the fundamental properties that define the system.