---- Let $\text{GL}_{n}$ denote the [[general linear group]] over a [[field|field]] $\mathbb{F}$ (we assume $\mathbb{F}=\mathbb{C}$ whenever necessary). > [!definition] Definition. ([[Lie group representation]]) > A **representation** of a [[Lie group]] $G$ is a [[group representation|representation of]] $G$ whose defining [[group homomorphism|homomorphism]] $\rho:G \to \text{GL}_{n}$ is [[smooth]], i.e. a [[Lie group|Lie group homomorphism]] $\rho:G \to \operatorname{GL}_{n}$. ^definition > [!NOTE] Note. > It is convenient to work over $\mathbb{C}$ and only as a secondary issue discuss whether a [[group representation|representation]] $(\rho, V)$ is real. The way 'view $\rho$ as real' is to > 1. Put a [[real structure]] $\sigma$ on $V$ > 2. Show that $\sigma$ commutes with the action[^4]: for all $g \in G$, $v \in V^{\sigma}$, $\sigma(g \cdot v)=g \cdot \sigma(v)$. > > An example can be found in B.1.b of [[LG Sheet 4]]. [^4]: The reason is that: for $(\rho, V)$ to be a 'real' representation, its realification (a $2\text{dim}_{\mathbb{C}}V$-dimensional $\mathbb{R}$-representation) needs to carry the $\mathbb{R}$-vector space $V^{\sigma}$ as a [[group subrepresentation|subrepresentation]]. For any $g \in G$, $v \in V^{\sigma}$ (so $v=\sigma(v)$), have $\begin{align} g \cdot v \in V^{\sigma} & \iff \sigma (g \cdot v)=g \cdot v \\ &\iff \sigma (g \cdot v) = g \cdot \sigma (v) \end{align}$which is where the intertwining condition came from. > [!basicexample] > - The **trivial representation** of $G$ is $g \mapsto \id$ for all $g \in G$. > - If $G$ is a [[Lie subgroup]] of $\operatorname{GL}(n, \mathbb{F})$, then the [[inclusion map|inclusion]] $G \hookrightarrow \operatorname{GL}(n, \mathbb{F})$ is a [[Lie group|Lie group homomorphism]], hence a [[Lie group representation]] of $G$. We call this the **defining representation** of $G$. ^basic-example $\rho$ gives rise to a natural [[linear map|linear]] [[group action|action]] $G \curvearrowright \{ \text{functions on }V \}$ as[^1] $g \cdot f:= f \circ \rho_{g ^{-1}}.$ #### The representation $U_{n}$ of $G= \operatorname{U}(1)=\mathbb{S}^{1}$ Any $n \in \mathbb{Z}$ defines a representation $\mathbb{S}^{1} \to \operatorname{Aut}(\mathbb{C})\cong\mathbb{C}^{\times}$ as $e^{i \theta} \cdot z := e^{i n \theta} z.$ We write **$U_{n}$** to denote this representation. Being one-dimensional, it is [[irreducible group representation|irreducible]]. ($n=0$ is the [[trivial group representation|trivial representation]].) In fact, this exhausts all $\mathbb{S}^{1}$-irreps: [[characterizing irreps of abelian groups|see here]]. #### The representation $V_{n}$ of $G=\operatorname{SU}(2)\cong\mathbb{S}^{3}$ The [[regular representation on functions]] $\mathbb{C}^{2} \to \mathbb{C}$ [[regular representation on functions|induced by the defining representation]] $\operatorname{SU}(2) \hookrightarrow \operatorname{GL}(2, \mathbb{C})$, $\begin{align} \operatorname{SU}(2) & \overset{\rho}{\curvearrowright}{}\{ \text{functions }\mathbb{C}^{2} \to \mathbb{C} \} \\ Q \cdot f & \mapsto f \circ \overline{Q}^{\top}, \end{align}$ restricts to [[symmetric power|a]] [[group subrepresentation|subrepresentation]][^2] $\operatorname{SU}(2) \curvearrowright \mathbb{C}[Z_{1}, Z_{2}]_{n}\cong \operatorname{Sym}^{n}(\mathbb{C}^{2})$ for any $n \in \mathbb{Z}$. We write **$V_{n}$** to denote this representation. It is [[irreducible group representation|irreducible]], although it's an exercise to show this (page 42). In fact, $V_{n}$ comprise *all* the [[irreducible group representation|irreps]], (essentially because $\operatorname{SU}(2)$ is [[compact]], [[simply connected]] ([[simple Lie group|simple too]]) and so its irreps are those of $\mathfrak{sl}_{2}(\mathbb{C})=\mathfrak{su}_{2} \otimes \mathbb{C}$ [[classification of the irreps of sl2 over C|which we have classified]]. This won't be on the exam, though). Can show too using [[representation ring of a Lie group|rep ring]] (that *is* valid for exam, and somewhat quicker).[^5] [^5]: Both approaches have some aspect that seems nonelementary. Classifying $\mathfrak{sl}_{2}(\mathbb{C})$-irreps is elementary (though a bit involved; it took us a whole lecture in Part III), but a priori one needs the [[the category of simply connected Lie groups is equivalent to that of finite-dimensional real lie algebras|Lie category equivalence]] (really, [[Lie's Second Theorem on integrating Lie morphisms|Lie II]]) to use this in classifying $\operatorname{SU}(2)$-irreps. Meanwhile, the representation ring approach a priori needs the fact that $R(G)\cong R(T)^{W}$ [[ring isomorphism|as]] [[ring|rings]]. However for the special case $\operatorname{SU}(2)$ both of these facts can be done manually without much trouble (e.g. surjectivity of the ring map can be manually checked). > [!justification] > We have to show that > 1. $\mathbb{C}[Z_{1},Z_{2}]_{n}$ is indeed [[group-invariant subspace|invariant]] under the action of $\operatorname{SU}(2)$... > 2. $V_{n}$ is irreducible ^justification ---- #### [^1]: The inverse is required as without it we don't get a genuine (left) [[group action]] (associativity will fail). [^2]: $\mathbb{C}[Z_{1},Z_{2}]_{n}$ denotes the [[vector space]] of [[homogeneous polynomial|homogeneous]] [[polynomial 4|polynomials]] [[graded ring|of degree]] $n$ over $\mathbb{C}$. ---- #### 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 > ```