Group Representations

Definition: Let $G$ be a group. A Group Representation of $G$ is a pair $(V, \rho)$ where $V$ is a vector space and $\rho : G \to \mathrm{GL}(V)$ is a group homomorphism from $G$ to $\mathrm{GL}(V)$.

Here, $\mathrm{GL}(V)$ denotes the group of all invertible linear maps from $V$ to $V$.

Sometimes it will be convenient to simply say that "$V$" is the group representation of $G$ when ambiguity won't arise.

If $(V, \rho)$ is a group representation of the group $G$ then define a group action $\cdot$ of $G$ on $V$ by:

(1)

\begin{align} \quad g \cdot v = [\rho(g)](v) \end{align}

Indeed, the above operation is a group action of $G$ on $V$ since for all $g_1, g_2 \in G$ and all $v \in V$ we have that since $\rho : G \to \mathrm{GL}(V)$ is a homomorphism:

(2)

\begin{align} \quad g_1 \cdot (g_2 \cdot v) &= [\rho(g_1)]([\rho(g_2)](v)) = [\rho(g_1) \circ \rho(g_2)](v) = [\rho(g_1g_2)](v) = (g_1g_2) \cdot v \\ \quad e \cdot v &= [\rho(e)](v) = [\mathrm{id}_V](v) = v \end{align}

Moreover, for each $g \in G$, for all $v_1, v_2, v \in V$, and for all $\lambda \in \mathbb{C}$ we have that:

(3)

\begin{align} \quad g \cdot (v_1 + v_2) &= [\rho(g)](v_1 + v_2) = [\rho(g)](v_1) + [\rho(g)](v_2) = g \cdot v_1 + g \cdot v_2 \\ \quad g \cdot (\lambda v) &= [\rho(g)](\lambda v) = \lambda [\rho(g)](v) = \lambda (g \cdot v) \end{align}

Definition: Let $G$ be a group. A Finite-Dimensional Group Representation of $G$ is a group representation $(V, \rho)$ for which $\mathrm{dim}(V) < \infty$. An Infinite-Dimensional Group Representation of $G$ is a group representation $(V, \rho)$ for which $\mathrm{dim}(V) = \infty$.