Representations

A (complex) representation of $G$ is a homomorphism $\rho : G\to GL(V)$ with $V$ a finite-dimensional complex vector space; $\dim V$ is its degree. Choosing a basis gives matrices $\rho(g)\in GL_n(\mathbb{C})$ with $\rho(gh) = \rho(g)\rho(h)$. Equivalently, $G$ acts on $V$ by linear maps. Examples: trivial $\rho(g) = 1$; sign of $S_n$; permutation representation of $S_n$ on $\mathbb{C}^n$ by permuting coordinates; regular representation on $\mathbb{C}G$ (basis $e_h$, $g\cdot e_h = e_{gh}$); $C_n = \langle g\rangle$ with $\rho(g) = e^{2\pi ik/n}$.

A subspace $W\le V$ is $G$-invariant if $\rho(g)W\subseteq W$ for all $g$; it is then a subrepresentation. $V$ is irreducible if $V\ne0$ and its only invariant subspaces are $0$ and $V$. $V\oplus W$ is a representation via block-diagonal matrices. Representations $\rho, \rho'$ are equivalent ($\cong$) if $\rho'(g) = T\rho(g)T^{-1}$ for a fixed invertible $T$ (a change of basis intertwining them). Example: in the permutation representation of $S_n$, the line $\langle(1, \dots, 1)\rangle$ and the hyperplane $\{\sum x_i = 0\}$ are invariant.

$D_{2n} = \langle r, s\mid r^n = s^2 = e, srs^{-1} = r^{-1}\rangle$ acts on $\mathbb{C}^2$ by

For $n\ge3$ it is irreducible: an invariant line would be an eigenline of $\rho(r)$, whose eigenvectors $(1, \mp i)$ are swapped by $\rho(s)$.

A $G$-homomorphism $\phi : V\to W$ is a linear map with $\phi\rho_V(g) = \rho_W(g)\phi$ for all $g$. Its kernel is a subrepresentation of $V$ and its image a subrepresentation of $W$; an invertible $G$-homomorphism is exactly an equivalence. $\operatorname{Hom}_G(V, W)$ is a vector space.