Complex irreducible representation (reviewed)

A representation of a finite group $G$ is a homomorphism $\rho : G \to \GL(V)$ for some vector space $V$. A representation is reducible if there is some nontrivial subspace $W \subset V$ with $\rho(g)(W) = W$ for all $g \in G$; otherwise it is irreducible. We say two representations $\rho : G \to \GL(V)$ and $\rho' : G \to \GL(V')$ are equivalent if there is an isomorphism $\sigma : V \to V'$ so that $\rho'(g) = \sigma^{-1} \rho(g) \sigma$ for all $g \in G$. Up to equivalence, the complex irreducible representations of a finite group $G$ (also known as irreps) are determined by their characters, which are recorded in the character table of $G$.