----- > [!proposition] Proposition. ([[characterizing irreps of abelian groups]]) > **1.** For $G$ an [[abelian group]] [[group action|acting on]] a finite-dimensional $\mathbb{C}-$[[vector space]] $V$ via [[group representation|representation]] $(\rho, V)$, we have that $\rho$ is [[irreducible group representation|irreducible]] if and only if $\dim \rho = 1$. > > If $G$ is a [[compact]] [[connected]] [[abelian group|abelian]] [[Lie group]] ([[classification of connected abelian Lie groups|that is]], a torus) and $V$ is a [[Lie group representation]] of $G$, we can say more: > > **2.** If $V$ is complex, then $V$ is irreducible iff the restriction to each torus factor $\mathbb{S}^{1}$ identifies with $U_{n}$ for some $n \in \mathbb{Z}$, where $U_{n}$ is the representation $\big(\operatorname{U}(1) \to \operatorname{\operatorname{GL}(\mathbb{C})}\big) \cong \big( \mathbb{S}^{1} \to \mathbb{C}^{\times} \big)$ given by $e^{i \theta} \cdot z \mapsto e ^{ i \theta n} z$for any $n \in \mathbb{Z}$. ($U_{0}$ is the trivial complex representation.) > > **3.** If $V$ is real, then $V$ is irreducible if and only if one of the following holds for each restriction to a torus factor $\mathbb{S}^{1}$: > a) $V=\mathbb{R}$ is the [[trivial group representation|trivial representation]] > b) $V=\mathbb{R}^{2}$ is the $W_{n}$ of $\operatorname{SO}(2) \cong \operatorname{U}(1)$, $n \geq 1$, given by the [[restriction of scalars|realification]] $(U_{n})_{\mathbb{R}}$ thus: $e ^{i \theta} \cdot \begin{bmatrix} > v_{1} \\ v_{2} > \end{bmatrix} \mapsto \begin{bmatrix} > \cos (n \theta) & - \sin(n \theta) \\ > \sin(n \theta) & \cos(n \theta) > \end{bmatrix} \begin{bmatrix} > v_{1} \\ v_{2} > \end{bmatrix}$ > > > [!proof]- Proof. ([[characterizing irreps of abelian groups]]) > $\to$. 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. **1.** Can also just use [[Schur's lemma for groups|Schur's Lemma]]. Consider a $\mathbb{C}$-[[vector space]] $V$ that is an irrep of $G$. Observe that since $G$ is [[abelian group|abelian]], $\rho_{g}:V \to V$ is a [[morphism of group representations|morphism]] of irreps, hence for any $v \in V-\{ 0 \}$ we have $\rho_{g}v=\lambda v$ for some $\lambda \in \mathbb{C}-\{ 0 \}$. Now $\mathbb{C}\langle v \rangle$ is a nonzero [[group subrepresentation|subrep]] of $V$; conclude $\mathbb{C}\langle v \rangle=V$. Thus $\text{dim }V=1$. ----- #### ---- #### 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 > ```