---- > [!definition] Definition. ([[group-invariant subspace]]) > Let $\rho$ be a [[group representation|representation]] of a finite [[group]] $G$ on a [[vector space]] $V$. A [[linear subspace]] $W \subset V$ is called **$G$-invariant** if the [[group action|action]] restricts to $W$: $\rho_{g} [W] \subset W \text{ for all } g \in G,$ > \ > (Note that this implies $\rho_{g}[W]=W$, like how we defined normal subgroup to be $gNg ^{-1} \subset N$ but in fact $gNg ^{-1}=N$ since $\rho_{g}$ is a [[bijection]]). ---- #### ---- #### 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 > ```