Unitary representations of a group

Unitary representations of a group are mathematical ways to describe how the elements of a group can be represented as symmetrical transformations, particularly as complex matrices that preserve length and angles. This means that when a group’s action is applied to a vector in a space, the resulting vector maintains its original shape. These representations are important in physics and mathematics, particularly in quantum mechanics, where they help to understand symmetries in physical systems. Essentially, they provide a powerful framework for studying how groups can act on spaces without altering fundamental properties.