----- > [!proposition] Proposition. ([[character is irreducible iff has unit length]]) > The [[character of a representation|character]] $\chi$ of a [[group representation|representation]] $\rho$ on [[group]] $G$ is [[irreducible group representation|irreducible]] iff $\langle \chi,\chi \rangle=1$ wrt the [[inner product]] defined on the space of [[class function]]s on $G$. > [!proof]- Proof. ([[character is irreducible iff has unit length]]) > Suppose $G$ has $h$ [[conjugate|conjugacy classes]]. [[Maschke's Theorem|Then]] $\begin{align} > \rho = & \bigoplus_{i=1}^{h}m_{i} \rho_{i} \\ > \chi_{\rho} = & \sum_{i=1}^{h} m_{i} \chi_{i}, > \end{align}$ > where the $\rho_{i}$ are the irreps and $m_{i}=\langle \chi_{\rho}, \chi_{i} \rangle$ due to [[orthonormal basis|orthonormality]]. > > If $\langle \chi_{\rho}, \chi_{\rho} \rangle=1$, then this means $\left\langle \sum_{i=1}^{h}m_{i} \chi_{i}, \sum_{j=1}^{h}m_{j} \chi_{j} \right\rangle = \sum_{i,j} m_{i}m_{j} \langle \chi_{i}, \chi_{j} \rangle = \sum_{i} m_{i}^{2}= 1.$ > The $m_{i}$ are all natural numbers, hence if $\sum_{i}m_{i}^{2}=1$ that means only one term in the [[linear combination]] $\chi_{\rho}=\sum_{i=1}^{h} m_{i} \chi_{i}$ is nonzero with coefficient 1, so indeed $\rho_{i}=\chi_{i}$ for some $i \in [h]$. > > The converse is directly by [[class function|(ortho)normality]] of irrep characters, of course. ----- #### ---- #### 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 > ```