----- > [!proposition] Proposition. ([[representation of abelian group is irreducible iff it's 1-dimensional]]) > For $G$ a finite [[abelian group]] [[group action|acting on]] a finite-dimensional [[vector space]] $V$ via [[group representation|representation]] $(\rho, V)$, then $\rho$ is [[irreducible group representation|irreducible]] if and only $\dim \rho = 1$. > [!proof]- Proof. ([[representation of abelian group is irreducible iff it's 1-dimensional]]) > $\to$. If $\dim \rho = 1$ then immediate from definition it is [[irreducible group representation|irreducible]]. \ $\leftarrow$. Suppose $G$ is [[abelian group|abelian]]. Then the [[linear operator|linear operators]] $\{ \rho_{g} \}_{g \in G}$ are [[simultaneously diagonalizable]], by [[representation of finite abelian group consists of simultaneously diagonalizable operators]]. Using this, pick an [[eigenbasis]] of $V$ with respect to which all the $\mathcal{M}_{\rho _g}$ are [[diagonal]]. Each element in this [[basis]] [[submodule generated by a subset|spans]] a $G$-[[group-invariant subspace|invariant line]]. So $(\rho, V)$ can only be [[irreducible group representation|irreducible]] if it is one-dimensional. ----- #### ---- #### 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 > ```