----- > [!proposition] Proposition. ([[pseudo-orthogonality of matrix coefficients of nonisomorphic irreps]]) > Let $G$ be a finite [[group]] with complex [[irreducible group representation|irreducible ]] [[group representation|group representations]] $(\rho^{(1)}, V_{1}) \not \cong (\rho^{(2)}, V_{2})$. Fix [[basis|bases]] $\mathfrak{B}_{1}$, $\mathfrak{B}_{2}$ for $V_{1},V_{2}$; let $P(g)$ be the [[matrix]] of $\rho^{(1)}_{g}$ wrt $\mathfrak{B}_{1}$ and $Q(g)$ the [[matrix]] of $\rho^{(2)}_{g}$ wrt $\mathfrak{B}_{2}$. We have the following 'entrywise orthogonality' when averaging over $G$: for all $(i,i')$, $(j,j')$, where $1 \leq i,i' \leq m$ and $1 \leq j,j' \leq n$, $\frac{1}{|G|}\sum_{g \in G}q_{ii'}(g^{-1})p_{j'j}(g)=0.$ ^6713a8 > [!proof]- Proof. ([[pseudo-orthogonality of matrix coefficients of nonisomorphic irreps]]) > Let $A=e_{i'j'} \in \mathbb{C}^{m \times n}$ be the [[matrix]] with $1$ in the $i',j'$ entry and $0$ elsewhere. By [[Schur's lemma for groups]], the $G$-[[group-equivariant map|equivariant]] [[matrix]] $\tilde{A}$ formed by [[averaging over a group|averaging]] $A$ over $G$ equals $0$. However, the $(i,j)th$ entry of $\tilde{A}$ is exactly the sum $\frac{1}{|G|}\sum_{g \in G} q_{ii'}(g^{-1})p_{j'j}(g).$ So this sum is zero. ----- #### ---- #### 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 > ```