---- Let $V$ be a finite-dimensional [[vector space]] over $\mathbb{C}$. > [!definition] Definition. ([[character of a representation]]) > The **character** of a [[group representation|representation]] $(\rho, V)$ is the complex-valued function $\begin{align} \chi: G & \to \text{GL}(V) \\ g & \mapsto \text{tr } \rho_{g}, \end{align}$ where $\text{tr}$ denotes the [[trace of a linear operator|trace]] of the [[linear operator|operator]] $\rho_{g}$. \ The **dimension of $\chi$** is said to be that of $\rho$. > [!note] Note. > We often abuse notation and talk about $\chi_{\rho}$ is if it were $\rho$ itself. For example, the notation $\ker \chi_{\rho}$ refers to $\ker \rho$: $\ker \chi_{\rho} = \{ g \in G : \rho_{g} = I \}=\textcolor{Thistle}{\{ g \in G: \chi_{\rho}(g)= \chi_{\rho}(1) = \dim \rho \}}.$ Note that (one inclusion for) the $\text{\textcolor{Thistle}{final equality}}$ is not obvious. [^1] [^1]: A proof that $\ker \rho = \{ g \in G : \chi(g)=\chi(1) \}$, is as follows. Let $g \in \ker \rho$, so that $\rho_{g}=I$. Then $\chi(g)=\text{tr }I=\chi_{\rho}(1)$. So it's clear that $\ker \rho \subset \{ g \in G : \chi(g)=\chi(1) \}$. Conversely let $g \in G$ s.t. $\chi(g)=\chi(1)$. We want to show that $\underbrace{\sum_{i=1}^{\dim \rho} \lambda_{i}(\rho_{g})}_{=\text{tr } \rho_{g}= \chi(g)}= \dim \rho \iff \underbrace{\lambda_{i}(\rho_{g}) \equiv 1}_{{(\iff \rho_{g}=I)}},$ since $\rho_{g}$ is [[diagonalizable]] (all [[matrices of finite order in GL_n(C) are diagonalizable]]) and [[diagonalizable matrix has eigenvalues all 1 iff is identity]]. $\leftarrow$ is obvious; conversely suppose $\sum_{i=1}^{\dim \rho} \lambda_{i}(\rho_{g})= \dim \rho$. Since [[group homomorphisms preserve structure|we know]] $\rho_{g}^{|g|}=I$, [[eigenvalues of order-m matrix in GL_n(C) are mth roots of unity|it follows that]] $\text{eig}(\rho_{g}) \subset \{ \omega_{|g|}^{\ell} : \ell=0,\dots,|g|-1\}$, where $\omega_{|g|}$ is a $|g|^{th}$ [[roots of unity|root of unity]]. So $\Re \lambda_{i} \leq 1$ with equality iff $\lambda_{i}=1$. Since the sum $\dim \rho$ is real, all imaginary components will sum to 0, but then in order to sum to $\dim \rho$ we in fact need $\lambda_{i} \equiv \rho_{g}$. > [!basicproperties] > - $\chi_{\rho}(1)$ gives the dimension of the character $\chi$. > - $\chi_{\rho}$ is a [[class function]]: it is constant on [[conjugate|conjugacy classes]]. > - [[complex group representations are isomorphic iff their characters match]]. > - $\chi_{\rho}(g^{-1})= \overline{\chi_{\rho}(g)}$. > - If $|g|=d$, then $\chi_{\rho}(g)=\sum_{i=1}^{n} \zeta_{i}$, where each $\zeta_{i}$ is some $d^{th}$ [[root of unity]]. > - > [!basicproperties] One-Dimensional Characters. > > - A one-dimensional [[character of a representation|character]] $G \xrightarrow{\chi} \mathbb{C}^{\times}$ is a [[group homomorphism|homomorphism]], because $\chi(gh)=\rho_{gh}=\rho_{g}\rho_{h}=\chi(g)\chi(h)$. > > - Since $\mathbb{C}^{\times}$ is [[abelian group|abelian]], for such $\chi$ [[universal property of commutator subgroup|we have]] $[G,G] \leq \ker \chi$, where $[G,G]$ is the [[commutator subgroup]] of $G$. > > - $\chi$ is a [[class function]], so its [[kernel of a group homomorphism|kernel]] is the union of [[conjugate|conjugacy classes]] $C_{g}$ such that $\chi(g)=1$. > > > > > [!warning] > > > A [[character of a representation|character]] of dimension greater than $1$ is *not* a [[group homomorphism|homomorphism]]. We still refer to its 'kernel' sometimes. > ^b39c84 > > [!proof] Proof of Basic Properties. > **1.** Since $\rho$ is a [[group homomorphism|homomorphism]], $\chi_{\rho}(1)= \text{tr }I_{n}=n$ where $n$ is the dimension of the [[group representation|representation]] $\rho$. \ **2.** Given $x,y \in G$ and [[group representation|representation]] $\rho: G \to \text{GL}_{n}(\mathbb{C})$, $\begin{align} \chi_{\rho}(xyx ^{-1}) = & \text{tr } \rho_{xyx ^{-1}}= \text{tr } \rho_{y}, \end{align}$\ where we use the fact that the [[trace of a matrix]] is invariant under [[change of basis formula|conjugation of that matrix]] (as sum of [[eigenvalue|eigenvalues]] which are intrinsic to the operator). That is, $\chi$ is a [[class function]]. > > **4.** By [[eigenvalues of order-m matrix in GL_n(C) are mth roots of unity]] we know that for all $g \in G$, $\rho_{g}$ has as [[eigenvalue|eigenvalues]] $|g|^{th}$ [[roots of unity]]. Then compute $\chi_{\rho}(g^{-1})= \sum_{i} \lambda_{i}(\rho_{g^{-1}})= \sum_{i} \lambda_{i}^{-1}(\rho_{g}) = \sum_{i} \lambda_{i}^{*}=\overline{\chi_{\rho}(g)},$ where we've used [[eigenvalues of matrix power]] (power=-1). > **5.** As in 4, we know $\rho_{g}$ has as its $n$ [[eigenvalue]]s $d^{th}$ [[roots of unity]]. [[trace of a linear operator|Its trace]] is the sum of these. > [!basicexample] Characters of Representations of $S_{3}$. Let $S_{3}=\{ 1,x,x^{2},y, xy,x^{2}y \}$ denote the [[symmetric group]] on $3$ letters. [[group representation#^2b0605|recall that we have the following 3 representations]] of $S_{3}$: \ >- The [[trivial group representation]] $\mathbb{1}$ (1-dimensional); >- The [[sign representation]] $\Sigma$ (1-dimensional); >- The [[standard representation of the dihedral group|standard representation]] $A$ of $D_{3} \cong S_{3}$ (2-dimensional). \ Clearly $\chi_{\mathbb{1}}(\sigma) \equiv 1$ for all $\sigma \in S_{3}$. \ $\chi_{\Sigma} (\sigma)=1$ for the even permutations $\sigma \in \{ 1,x^{},x^{2} \}$ and $\chi_{\Sigma}=-1$ for the [[parity of a permutation|odd permutations]] $\sigma \in \{ y,xy,x^{2}y \}$. \ The reflections are [[conjugate]] in $S_{3}$ and [[trace of a matrix|trace]] is invariant under [[change of basis formula|matrix conjugation]], hence $\chi_{A}(r)=\text{tr } \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}=0$ for $r \in \{y, xy, x^{2}y \}$. For the [[conjugate]] rotation matrices it suffices to consider $\text{tr }\begin{bmatrix} \cos \frac{2\pi}{3} & -\sin \frac{2\pi}{3}\\ \sin \frac{2\pi}{3} & \cos \frac{2\pi}{3} \end{bmatrix}=2\cos \frac{2\pi}{3}=2\left( -\frac{1}{2} \right)=-1.$ So $\chi_{A}(\text{rot})=-1$ for $\text{rot} \in \{ x,x^{2} \}$. Compiling as a table we get > | | $1$ | $x, x^2$ | $y, xy, x^2y$ | |----------|-----|----------|---------------| | $\chi _\mathbb{1}$ | $1$ | $1$ | $1$ | | $\chi _\Sigma$ | $1$ | $1$ | $-1$ | | $\chi _A$ | $2$ | $-1$ | $0$ | > The columns of the [[character table]] are [[orthogonal]]. The rows weighted by [[conjugate|conjugacy class size]] are as well. This is not a coincidence: see [[class function|the orthogonality relations]]. . ^9ecc4a ---- #### ---- #### 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 > ```