---- > [!definition] Definition. ([[group-averaged Hermitian form]]) > Let $G$ be a finite [[group]] [[group action|acting on]] a [[finite-dimensional vector spaces MOC|finite-dimensional]] [[complex numbers]]- [[inner product space|inner product space]] [[inner product space]] via [[group representation|representation]] $(\rho, V)$. The [[inner product]] $\langle \langle v_{1},v_{2} \rangle \rangle := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g) v_{1}, \rho(g)v_{2} \rangle ,$ > obtained by [[averaging over a group|averaging]] $\langle \cdot, \cdot\rangle$ over $G$, is called a **$G$-averaged Hermitian form** on $V$. > \ > Crucially, $\langle \langle \cdot,\cdot \rangle \rangle$ is [[group-invariant function|G-invariant]]. ---- #### ---- #### 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 > ```