---- Let $G$ be a finite [[group]] and $V$ a [[vector space]] over $\mathbb{C}$. > [!definition] Definition. ([[morphism of group representations]]) > An **morphism** from one $G$-[[group representation|representation]] $(\rho, V)$ to another $(\rho',V')$ is a [[linear map|linear map]] $T: V \to V'$ that is $G$-[[group-equivariant map|equivariant]], i.e., that commutes with the [[group action]]: $T(g \cdot v)= g \cdot T(v).$ If $T$ is [[bijection|bijective]], we call it an **isomorphism of representations**. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amssymb} > > \begin{document} > > \begin{tikzcd} > V \arrow[r, "T"] \arrow[d, "\rho_g"'] & V' \arrow[d, "\rho'_g"] \\ > V \arrow[r, "T"] & V' > \arrow[from=1-1, to=2-2, "\circlearrowleft", phantom] > \end{tikzcd} > > \end{document} > ``` ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```