---- > [!definition] Definition. ([[tensor product of group representations]]) > Let $G$ be a [[group]] and $V,W$ two $G$-[[group representation|representations]]. Their [[tensor product of modules|tensor product]] $V \otimes W = \span \{ v \otimes w \}_{v \in W, w \in W}$ is a $G$-representation by linear extension of the [[group action|action]] $g \cdot (v \otimes w)=(g \cdot v) \otimes (g \cdot w).$ ^definition > [!basicexample] > One has $V^{\oplus m} \cong V \otimes \mathbb{C}^{m}$. ^basic-example Compare: ![[tensor product of Lie algebra representations]] ---- #### ---- #### 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 > ```