---- Let $G$ be a finite [[group]] and $\rho_{}^{(1)}$ resp. $\rho_{}^{(2)}$ [[group representation|representations]] of $G$ on [[vector space|vector spaces]] $V_{1}$ resp. $V_{2}$. > [!definition] Definition. ([[direct sum of representations]]) > The **external direct sum of representations** $(\rho_{}^{(1)}, V_{1}) \oplus (\rho^{(2)}, V_{2})$ is defined as the [[group representation|representation]] of $G$ on the [[direct sum|external direct sum]] $V_{1} \oplus V_{2}:=\{ (v_{1},v_{2}): v_{1} \in V_{1}, v_{2} \in V_{2} \}$ obtained via the defining [[group homomorphism|homomorphism]] $\begin{align} \\ \rho: G & \to \text{GL}(V_{1} \oplus V_{2}) \\ g & \mapsto \underbrace{(\rho_{g}^{(1)}, \rho^{(2)_{}}_{g})}_{{\rho_{g}}} \\ v = (v_{1},v_{2}) & \xmapsto{\rho_{g}} \big([](direct%20sum%20of%20vector%20spaces.md){g}(v_{2})\big). \end{align}$ \ Let $V$ be a [[vector space]] that is a [[direct sum]] of [[group-invariant subspace|G-invariant subspaces]], $V=W_{1} \oplus W_{2}$ [^1]. Defining $\alpha, \beta$ as restrictions of $\rho$ to $W_{1}$, $W_{2}$ respectively, we write $\rho=\alpha \oplus \beta$ and call [[group representation|representation]] $\rho$ the **internal direct sum of representations** $\alpha$ and $\beta$. Given [[basis|bases]] $B_{1}$ and $B_{2}$ for $W_{1}$ and $W_{2}$, $(B_{1},B_{2})$ is a [[basis]] for $V$, with respect to which the [[matrix]] of $\rho_{g}$ is [[block matrix|block diagonal]]: $\mathcal{M}_{\rho_{g}}=\begin{bmatrix} \mathcal{M}_{\alpha_{g}} & \b 0 \\ \b 0[](direct%20sum%20of%20vector%20spaces.md)}_{\beta_{g}} \end{bmatrix},$ due to the [[invariant subspace|invariance of]] $W_{1}$ under $\alpha$ and $W_{2}$ under $\beta$. Conversely, given a [[block matrix|block diagonal]] matrix representation (with square blocks) the corresponding representation is a direct sum. [^1]: We only care about *$G$-invariant* subspaces because $G$ does not even [[group action|act on]] subspaces $W$ which are not $G$-invariant (in which case it makes no sense to talk about a [[group representation|representation]] of $G$ on $W$). ---- #### ---- #### 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 > ```