---- Let $V$ be a $1$-[[dimension|dimensional]] [[vector space]] over a [[field]] $\mathbb{F}$ (often $\mathbb{F}=\mathbb{C}$) so that $\text{GL}(V) \cong \text{GL}_{1}(\mathbb{C})\cong \mathbb{C}^{\times}$. > [!definition] Definition. ([[trivial group representation]]) > For $G$ a finite [[group]], the **trivial representation** of $G$ on $V$ the [[group representation|representation]] characterized by the [[group homomorphism|homomorphism]] $\begin{align} \rho: G & \to \mathbb{C}^{\times} \\ g & \mapsto 1. \end{align}$ **Remark.** Any [[group]] has a trivial [[group representation|representation]]. ---- #### ---- #### 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 > ```