----- > [!proposition] Proposition. ([[regular representation contains all irreducibles with multiplicity equal to their dimension]]) > Let $(\rho,V)$ be any [[irreducible group representation]] of a finite [[group]] $G$. Then $\rho$ occurs in the [[regular representation|regular]] [[group representation|representation]] $\rho^{\text{reg}}$ of $G$ with multiplicity equal to $\dim \rho$. > [!proof]- Proof. ([[regular representation contains all irreducibles with multiplicity equal to their dimension]]) > Let $\chi_{\rho}$ denote the [[character of a representation|character]] of $\rho$. Using [[class function|the orthogonality relations]] and supposing $G$ has $h$ [[conjugate|conjugacy classes]], we know that $\chi_{\rho}= \sum_{i=1}^{h} \langle \chi_{\rho}, \chi_{i}\rangle \rho_{i}$. By [[regular representation|regular representation properties]] we have $\langle \chi_{\rho}, \chi_{i}\rangle = \dim \rho_{i}$ for all $i$. This is the result. ----- #### ---- #### 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 > ```