Group representation

Summary

A representation of a group $G$ on a vector space $V$ is a group homomorphism $G \to GL(V)$. Here $GL(V)$ is the group of invertible linear maps $V \to V$. It lets each element of $G$ act on $V$ as a linear transformation, compatibly with the group operation. Representations turn questions about groups into questions of linear algebra, where matrices, traces and eigenvalues are available. They are central in group theory, number theory, physics and chemistry.