----- > [!proposition] Proposition. ([[matrix coefficients of a single irrep]]) > Let $G$ be a finite [[group]] acting on a [[vector space]] $V$ via [[irreducible group representation|irrep]] $(\rho, V)$. Pick a [[basis]] $\mathfrak{B}$ for $V$ and let $P(g)$ be the [[matrix]] of $\rho_{g}$ wrt this [[basis]]. Then $\frac{1}{|G|} \sum_{g \in G} p_{ii'}(g^{-1})p_{j'j}(g)=\frac{\delta_{ij}\delta_{i'j'}}{\dim V}.$ > ^21957b > [!proof]- Proof. ([[matrix coefficients of a single irrep]]) > Consider the [[matrix]] $A:=e_{i'j'}$ that has 1 in its $(i',j')^{th}$ entry and $0$ elsewhere. Note that $\text{tr }A=\delta_{i'j'}$, hence using the [[averaging over a group|special case described here]] $\tilde{A}=\frac{\text{tr }A}{\dim V} I = \frac{\delta_{i'j'}}{\dim V}I.$ But the $(i,j)^{th}$ entry of $\tilde{A}$ is $\frac{1}{|G|} \sum_{g \in G} p_{ii'}(g^{-1})p_{j'j}(g)$. The proposition 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 > ```