----- > [!proposition] Proposition. ([[character of regular representation]]) > The [[character of a representation|character]] of the [[regular representation]] $\rho^{\text{reg}}$ of a finite [[group]] $G$ is $\chi_{\text{reg}}(g)=\begin{cases} |G| & g=e \\ 0 & \text{otherwise.} \end{cases}$ > [!proof]- Proof. ([[character of regular representation]]) > The [[matrix|matrices]] $R^{\text{reg}}_{g}$ of $\rho^{\text{reg}}$ are [[permutation matrix|permutation matrices]]; thus their [[diagonal]] elements are either $1$ (if the corresponding group element is [[stabilizer|stabilized]] by $g$) or $0$ (otherwise). But the [[regular representation]]'s [[group action]] is [[transitive group action|transitive]], [[orbit-stabilizer theorem|so this will only happen]] when $g=e$. Taking the [[trace of a matrix|trace]] of $R_{g}^{\text{reg}}$ the result follows. ----- #### ---- #### 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 > ```