----- > [!proposition] Proposition. ([[order of group equals sum of squares of irrep dimensions]]) > Let $G$ be a finite [[group]] with $h$ [[conjugate|conjugacy classes]] $[g]$. [[number of irreps equals number of conjugacy classes|Then]] $g$ has, up to [[morphism of group representations|isomorphism]] $h$ distinct [[irreducible group representation|irreps]] $\{ \rho_{i} \}_{i=1}^{h}$ over [[complex numbers]] with [[complex group representations are isomorphic iff their characters match|corresponding]] [[character of a representation|characters]] $\{ \chi_{i} \}_{i=1}^{h}$. For all $g,g' \in G$, we have $\sum_{i=1}^{h} \overline{\chi_{i}(g)}\chi_{i}(g')= \begin{cases} 0 & [g] \neq [g']; \\ |Z_{G}(g)| & [g]=[g'], \end{cases}$ where $Z_{G}(g)$ denotes the [[centralizer of an element in a group|centralizer]] of $g$ in $G$. \ This is the 'orthogonal columns' part of the [[class function|orthogonality relations]]. > [!proposition] Corollary. > Specializing to $g=e=g'$, we see that $\sum_{i=1}^{h} \overline{\chi_{i}(e)} \chi_{i}(e) = \sum_{i=1}^{h} \dim^{2}\rho_{i}=|G|,$ > where $|G|$ is the [[order of a group|order]] of $G$. > \ > **Remark.** Once the [[class function|orthogonality relations]] have been proven, this corollary is also a corollary of [[regular representation contains all irreducibles with multiplicity equal to their dimension]]. > [!proof]- Proof. ([[order of group equals sum of squares of irrep dimensions]]) > Let us order the [[conjugate|conjugacy classes]] of $G$ as $[a_{1}],\dots,[a_{h}]$. Consider the $h \times h$ [[matrix]] $A$ whose $(i,j)^{th}$ entry is $a_{ij}=\chi_{i}([a_{j}]).$ Let $B$ be the modified [[matrix]] whose $(i,j)^{th}$ entry is $b_{ij}= a_{ij} \left( \frac{|[a_{ij}]|}{|G|} \right) ^{1/2}.$ The [[orthonormal basis|orthonormality]] of the [[character of a representation|characters]] as functions in $\text{Cl}(G)$ implies that the rows of $B$ are an [[orthonormal basis]] of $\mathbb{C}^{n}$. So $B$ is [[isometric matrix|unitary]], hence its columns are [[orthonormal]] too. Explicating this gives ![[CleanShot 2023-12-09 at 11.52.46.jpg]] ----- #### ---- #### 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 > ```