---- > [!definition] Definition. ([[class function]]) > A **class function** on a [[group]] $G$ is a function on $G$ that is constant on [[conjugate|conjugacy classes]]. > \ > The set of **class functions** $\text{Cl}(G)$ from a [[group]] $G$ to the [[complex numbers]] forms a [[vector space]] over $\mathbb{C}$ of [[dimension|dimension]] $\# \text{CCs}$. There is an [[inner product]] on this space defined by $\langle \phi, \psi \rangle= \frac{1}{|G|} \sum_{g \in G} \overline{\phi(g)} \psi(g) ,$ w.r.t. which the [[irreducible group representation|irreducible]] [[character of a representation|characters]] of $G$ form an [[orthonormal basis]]. > [!justification] > The [[orthonormal basis|orthonormality]] of [[character of a representation|characters]] is shown in the propositions below. That there are $\dim \text{Cl} (G)$ of them (and hence they indeed form an [[orthonormal basis]]) is shown by [[number of irreps equals number of conjugacy classes]] along with [[complex group representations are isomorphic iff their characters match]]. > [!proposition] Proposition. (Orthogonality of Irreducible Characters) > If $(\rho^{(1)},V_{1})$ and $(\rho^{(2)},V_{2})$ are [[irreducible group representation|irreducible]] and not [[morphism of group representations|isomorphic]], then $\frac{1}{|G|} \sum_{g \in G} \overline{\chi_{\rho^{(2)}}(g)} \chi_{\rho^{(2)}}(g)=0.$ > [!proof] Proof of Proposition. > Recall:![[pseudo-orthogonality of matrix coefficients of nonisomorphic irreps#^6713a8]] [[character of a representation|Noting that]] in general $\overline{\chi(g)}=\chi(g^{-1})$, we can compute $\begin{align} \frac{1}{|G|} \sum_{g \in G} \overline{\chi_{\rho^{(2)}}(g)} \chi_{\rho^{(2)}}(g) = & \frac{1}{|G|} \sum_{g \in G} {\chi_{\rho^{(2)}}(g^{-1})} \chi_{\rho^{(2)}}(g) \\ = & \frac{1}{|G|} \sum_{g \in G} \left[ \left( \sum_{i=1}^{m} q_{ii}(g^{-1}) \right) \left( \sum_{j=1}^{n} p_{jj} (g) \right) \right] \\ =& \frac{1}{|G|} \sum_{i,j} \frac{1}{|G|} \sum_{g \in G} q_{ii}(g^{-1}) p_{jj}(g) \\ =& 0, \text{ by the lemma}. \end{align}$ > [!proposition] Proposition. (Ortho*normality* of irreducible characters) > If $(\rho, V)$ is [[irreducible group representation|irreducible]], then $\frac{1}{|G|}\sum_{g \in G} \overline{\chi_{\rho}(g)} \chi_{\rho}(g)=1.$ > [!proof] Proof of Proposition. > Recall: ![[matrix coefficients of a single irrep#^21957b]] > Now we have $\begin{align} \frac{1}{|G|} \sum_{g \in G} \overline{\chi_{\rho}(g)} \chi_{\rho}(g) = & \frac{1}{|G|} \sum_{g \in G} \chi_{\rho}(g^{-1}) \chi_{\rho}(g) \\ = & \frac{1}{|G|} \sum_{g \in G} \left[ \left( \sum_{i} p_{ii}(g^{-1}) \right) \left( \sum_{j} p_{jj}(g) \right) \right] \\ = & \sum_{i,j} \underbrace{\frac{1}{|G|} \sum_{g \in G} p_{ii}(g^{-1})p_{jj}(g)}_{\frac{\delta_{ij}^{2}}{\dim V}} \\ = & \sum_{i,j} \frac{\delta _{ij}}{\dim V} \\ = & \dim V / \dim V \\ = & 1. \end{align}$ > [!proposition] Lemma. > Let $\mathcal{C}$ denote the subspace of the [[vector space]] $\mathcal{H}$ of [[class function]]s spanned by the [[irreducible group representation|irreducible]] [[character of a representation|characters]]. The [[orthogonal complement|orthogonal complement]] of $\mathcal{C}$ is $\{ 0 \}$, and thus the [[irreducible group representation|irreducible]] [[character of a representation|characters]] [[submodule generated by a subset]] $\mathcal{H}$. We prove this by showing: **a.** Let $\phi$ be a [[class function]] on $G$ that is [[orthogonal]] to every [[character of a representation|character]]. For any [[group representation|representation]] $\rho$ of $G$, $T=\frac{1}{|G|} \sum_{g} \overline{\phi(g)} \rho_{g}$ is the zero [[linear operator|operator]]. > [!proof] Proof of Lemma. > **Proof of a.** WLOG $\rho$ is [[irreducible group representation|irreducible]] (okay to do because of [[Maschke's Theorem]]). It is easy to show $T$ is [[group-equivariant map|G-equivariant]], Let $\chi$ be the [[character of a representation|character]] of $\rho$. The [[trace of a linear operator|trace]] of $T$ is $\text{tr }T = \frac{1}{|G|} \sum_{g} \text{tr } (\overline{\phi(g)}\rho_{g})=\frac{1}{|G|} \sum_{g} \overline{\phi(g)} \chi(g) = \langle \phi, \chi \rangle $ which equals $0$ by assumption that $\phi$ is [[orthogonal]] to all irreducible characters. By [[Schur's lemma for groups]], $T$ is multiplication by a [[scalar]], $T=cI$. But since $\text{tr }T = 0$, $c=0$ and thus $T=0$. ---- #### ---- #### 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 > ```