Isomorphisms of Group Representations

# Isomorphisms of Group Representations

Definition: Let $G$ be a group and let $(V_1, \rho_1)$ and $(V_2, \rho_2)$ be group representations of $G$. An Group Representation Isomorphism of these group representations if a function $\varphi : V_1 \to V_2$ that is a vector space isomorphism and is $G$-invariant, that is, for all $g \in G$ and for all $v_1 \in V_1$ we have that $\varphi([\rho_1(g)](v_1)) = [\rho_2(g)](\varphi(v_1))$. If such an isomorphism exists, we say that $(V_1, \rho_1)$ and $(V_2, \rho_2)$ are isomorphic group representations of $G$. |

Quite naturally - we can define an equivalence relation on the set of all group representations of a group $G$, and say that two group representations of $G$ are equivalent if and only if they isomorphic.

Our goal would be to be able to classify all group representations of a group $G$ up to isomorphism.

Proposition 1: Let $G$ be a group. Let $(V, \rho)$ be a group representation of $G$ and let $\varphi : V \to W$ is a vector space isomorphism. Let $\psi : G \to \mathrm{GL}(W)$ be defined for all $g \in G$ and for all $w \in W$ by $[\psi(g)](w) = [\varphi \circ \rho(g) \circ \varphi^{-1}](w)$. Then $(W, \psi)$ is a group representation of $G$ that is isomorphic to $(V, \rho)$. |

As a consequence of Proposition 1 we see that if $G$ is a group and $(V, \rho)$ is a finite-dimensional group representation of $G$ with $\mathrm{dim}(V) = n$ then we can choose a basis of $V$ and obtain an isomorphism $\varphi : V \to \mathbb{C}^n$ which will give us a representation $(\mathbb{C}^n, \rho')$ of $G$ that is isomorphism to $(V, \rho)$