📂Representation Theory

Intertwining Map

Definition¹

Let $G$ be a group, and let $(\rho, V)$ and $(\zeta, W)$ be its representations. A linear map $\phi : V \to W$ satisfying the following is called an intertwining map of $\rho$ and $\zeta$.

$$ \phi\left( \rho[g] v \right) = \zeta[g] \phi(v), \quad \forall g \in G,\enspace \forall v \in V $$

If $\phi$ is an invertible function, then $\phi$ is called an isomorphism, and the two representations $\rho$ and $\zeta$ are said to be isomorphic to each other.

• The same definition applies to representations of Lie algebras.

Explanation

In short, it suffices to understand that a linear equivariant map is specially called an intertwining map. If we simply denote the transformations $\rho(g)$ and $\zeta(g)$ by $g = \rho(g) = \zeta(g)$, it becomes clear that an intertwining map is a map that commutes with the action of the group $G$.

$$ g \phi(v) = \phi(gv) $$

Theorem

Let $G$ be a connected matrix Lie group. Let $\Pi_{1}$ and $\Pi_{2}$ be representations of $G$. Let $\pi_{1}$ and $\pi_{2}$ be representations of the Lie algebra of $G$. Then $\pi_{1}$ and $\pi_{2}$ are isomorphic if and only if $\Pi_{1}$ and $\Pi_{2}$ are isomorphic.

$$ \text{$\pi_{1}$ and $\pi_{2}$ are isomorphic $\iff$ $\Pi_{1}$ and $\Pi_{2}$ are isomorphic} $$