Definition
Proposition The space of -maps from to is exactly the -invariant subspace of the representation .
means the following:
\subsection*{1. What is a -map?}
Let and be vector spaces equipped with linear actions of a group , i.e., -representations via [ \rho_V: G \to \mathrm{GL}(V), \quad \rho_W: G \to \mathrm{GL}(W). ]
A linear map is called a \textbf{-map} (or -equivariant map) if it commutes with the action of , that is, [ f(\rho_V(g) v) = \rho_W(g) f(v), \quad \text{for all } g \in G,, v \in V. ] Equivalently, [ f \circ \rho_V(g) = \rho_W(g) \circ f \quad \text{for all } g \in G. ]
The space of all such maps is denoted by: [ \mathrm{Hom}_G(V, W). ]
\subsection*{2. The -action on }
The space of all linear maps from to , denoted , can itself be turned into a -representation. The group acts on by: [ (g \cdot T)(v) := g \cdot (T(g^{-1} \cdot v)) \quad \text{for } T \in \mathrm{Hom}(V, W),, g \in G,, v \in V. ]
This action is sometimes called the \emph{conjugation action}, and it ensures that the space is a -representation.
\subsection*{3. Characterizing -maps as -invariants}
A linear map is a -map if and only if it is fixed under the -action defined above: [ g \cdot T = T \quad \text{for all } g \in G. ]
Therefore, the space of -maps is precisely the -invariant subspace: [ \mathrm{Hom}_G(V, W) = \mathrm{Hom}(V, W)^G. ]
\subsection*{4. Conclusion}
The -equivariant maps from to correspond exactly to the -invariant elements of the representation under the conjugation action.
This perspective is extremely useful in representation theory, especially when studying morphisms between representations, and plays a key role in results like Schur’s Lemma.
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