----- > [!proposition] Proposition. ([[complex group representations are isomorphic iff their characters match]]) > Let $G$ be a finite [[group]] and $\rho,\rho'$ two [[group representation|representations]] of $G$. Then $\rho_{1}$ and $\rho_{2}$ are [[morphism of group representations|isomorphic]] if and only if their [[character of a representation|characters]] $\chi_{}$, $\chi_{}'$ are equal: $\rho \sim \rho' \iff \chi_{} = \chi_{}'.$ > > [!proof]- Proof. ([[complex group representations are isomorphic iff their characters match]]) $\to.$ One direction is obvious: suppose $\rho \sim \rho'$ via some [[linear isomorphism]] $T$. Then for any $g \in G$, $\chi(g)=\text{tr }\rho_{g} = \text{tr } T^{-1} \rho_{g}' T = \text{tr }\rho_{g}' = \chi'(g),$ where we used the fact that the [[eigenvalue|eigenvalues]], and therefore [[trace of a linear operator|trace]], of $\rho_{g}$ is invariant under [[conjugate|conjugation]]. > $\leftarrow.$ This is a consequence of [[class function|the orthogonality relations]]. Suppose $\rho,\rho'$ have the same [[character of a representation|character]] $\chi$. [[Maschke's Theorem|Then]] clearly $\rho \cong \bigoplus_{i=1}^{h} \langle \chi, \chi_{i} \rangle \chi_{i} \cong \rho '$ and we're done. ----- #### ---- #### 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 > ```