---- Let $D_{n}$ denote the [[dihedral group]] $\langle x,y | x^{n}=y^{2}=xyxy=e \rangle$. It can be [[dihedral group and linear isomorphisms of the plane|identified with]] the [[subgroup]] of $\text{GL}_{2}(\mathbb{\mathbb{C}})$ [[generating set of a group|generated by]] $O:=\begin{bmatrix} \cos \frac{2\pi}{n} & -\sin \frac{2\pi}{n} \\ \sin \frac{2\pi}{n} & \cos \frac{2\pi}{n} \end{bmatrix} \text{ and } R:= \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$ via the [[group isomorphism|isomorphism]] $\phi : D_{n} \to \text{GL}_{2}(\mathbb{C}), \ \ x \mapsto{^{}} O, y \mapsto R.$ > [!definition] Definition. ([[standard representation of the dihedral group]]) > The **standard representation of $D_{n}$** is the [[group representation|representation]] characterized by the [[group isomorphism|isomorphism]] $\phi$. It is [[faithful group representation|faithful]]. > [!note] Remark. > The analogy for [[Lie algebra|Lie algebras]] would be the [[defining representation of a Lie algebra|defining representation]]. ^note ---- #### ---- #### 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 > ```