---- Let $W$ be a [[vector space]] over a [[field]] $\mathbb{F}$ (often $\mathbb{F}=\mathbb{C}$). Let $G$ be a [[group]]. > [!definition] Definition. ([[trivial group representation]]) > The **trivial representation of $G$ on $W$** is the [[group homomorphism|homomorphism]] $\begin{align} G & \xrightarrow{1} \operatorname{GL}(W) \\ g & \mapsto \id_{W}. \end{align}$(The unadorned term "trivial representation" usually has $W=\mathbb{F}$ by default.) > > [!definition] For a general [[group representation|representation]] $(\rho, V)$ of $G$, we denote by $V^{G}$ the maximal trivial [[group subrepresentation|subrepresentation]] of $V$, i.e., $V^{G}:= \{ v \in V : g \cdot v \text{ for all } g \in G \}=:\text{FixedPoints}(\rho).$ > [!basicproperties] Properties of the MTS of $V$. > > 1. Can view $V^{G}\cong \text{Hom}_{G\mathsf{Rep}}(1, V)$ > 2. When $G$ is a [[compact]] [[Lie group]] (so admits [[Haar measure]] $\int_{G}$), $V^{G}=\left\{ \int_{G} \rho_{g}(w) : w \in V \right\}.$ > 3. As a corollary to the above, when $G$ is a [[compact]] [[Lie group]], $\underbrace{ \text{dim }V^{G} }_{ \text{rep-theoretic} } = \underbrace{ \int_{G} \chi_{V}(g) }_{ \text{analytic} },$ > where $\chi_{V}$ denotes the [[character of a representation|character]] of $V$. ---- #### [^1]: - [ ] todo ---- #### 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 > ```