---- Let $V$ be a finite-dimensional [[vector space]] over $\mathbb{C}$. $G$ is a [[compact]] [[Lie group]] (including the case that $G$ is finite). > [!definition] Definition. ([[character of a representation]]) > The **character** of a [[Lie group|(Lie)]] [[group]] [[group representation|representation]] $(\rho, V)$ is the complex-valued function $\begin{align} \chi: G & \to \mathbb{C} \\ g & \mapsto \text{tr } \rho_{g}, \end{align}$ 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 $\rho$. > [!basicexample] ^basic-example Throughout, $t:=e^{i \theta}$. - [[characterizing irreps of abelian groups|The repreentation]] $U_{n}$ of $\operatorname{U}(1)\cong \mathbb{S}^{1}$ has character $t \mapsto t^{n}$. Take $G=\operatorname{SU}(2)$ [[special unitary group|with]] [[irreducible group representation|irreps]] $V_{n}$. [[general orthogonal group|Restrict to]] $\operatorname{U}(1)$ via $\begin{align} \operatorname{U}(1) &\hookrightarrow \operatorname{SU}(2) \to \operatorname{GL}(V_{n}) \\ t & \mapsto \begin{bmatrix} t & \\ & \overline{t} \end{bmatrix}. \end{align}$ As a representation *of $\operatorname{U}(1)$*, $V_{n}$ [[direct sum of representations|splits as]][^3] $V_{n}\cong U_{n} \oplus U_{n-2} \oplus\dots \oplus U_{-n}.$ The character of the this representation is thus $t \mapsto t^{n}+t^{n-2}+\dots + t^{-n}=\sum_{j=0}^{2n}t^{-n+2j}.$ (Compare to the example in [[Weyl character formula]]?) - As a representation *of $\operatorname{SU}(2)$*, the character of $V_{n}$ is $g \mapsto t^{n}+t^{n-2}+\dots + t^{-n}.$ This is because WLOG $g$ is [[diagonal]] (characters are invariant under [[conjugate|conjugation]]/$\chi$ is a [[class function]] and the [[Complex Spectral Theorem]] exists), thus of the form $g=\begin{bmatrix}t & \\& \overline{t}\end{bmatrix}$ for some $t=e^{i \theta} \in \mathbb{S}^{1}$, and in the example/footnote above we saw that the map $g \cdot \_:V_{n} \to V_{n}$ had [[eigenvector|eigenvectors]] of the form $Z_{1}^{n-k}Z_{2}^{k}$ with [[eigenvalue|eigenvalues]] $t^{-n+2k}$. *i guess coming back to this, phrase in terms of character being determined by its restriction to the maximal torus so often that's really all we must consider, although careful to not be circular since we hadn't defined maximal tori yet.* Well i guess it's all the same but let me try to do this again more manually. Don't need strong result about [[maximal torus in a Lie group|maximal tori]] because in this case the [[Complex Spectral Theorem]] and $\chi$ being a [[class function]] means WLOG $g=\begin{bmatrix}t & \\& t ^{-1}\end{bmatrix}$ for some $t=e^{i \theta} \in \mathbb{S}^{1}$. Now $\begin{align} g \cdot \sum_{k=0}^{n} a_{k}(z_{1}^{n-k}z_{2}^{k}) &= \sum_{k=0}^{n} a_{k} \big( t^{-1} z_{1} \big)^{n-k} (t z _{2})^{k} \\ &= \sum_{k=0}^{n} a_{k} t^{-n+k} z_{1}^{n-k} t^{k }z_{2} ^{k} \\ &= \sum_{k=0}^{n} t^{-n+2k} a_{k} z_{1}^{n-k} z_{2}^{k} , \end{align}$ or in coordinates on $V_{n}=\span(z_{1}^{n}, z_{1}^{n-1}z_{2},\dots, z_{1}z_{2}^{n-1}, z_{2} ^{n})$, $[\boldsymbol \rho_{g}] \begin{bmatrix} a_{0}\\ \vdots \\ a_{n} \end{bmatrix} = \begin{bmatrix} t^{-n} \\ & t ^{-n+2} \\ & & \ddots \\ & & & t^{n} \end{bmatrix}\begin{bmatrix} a_{0}\\ \vdots \\ a_{n} \end{bmatrix}$ from which we see $\operatorname{tr}(\rho_{g})=t^{n}+ t^{n-2}+\dots + t^{-n}$. Equivalent but less writing is to recognize that the defining basis for $V_{n}$ is in fact an [[eigenbasis]] and compute trace as the sum of the eigenvalues which are precisely $t^{-n}$, $t^{-n+2},\dots$. This gets toward [[factorizing a torus representation into weight spaces|weights]]. [^3]: A way to see this is to first identify $\mathbb{C} Z_{1}^{n-k}Z_{2}^{k} \cong U_{n-2k}$, clearly they're isomorphic as vector spaces (one-dimensional), and ($n>k$ for now) $\begin{align} \begin{bmatrix} t & \\ & \overline{t} \end{bmatrix} \cdot Z_{1}^{n-k} Z_{2}^{k} &= (Z_{1} \overline{ t})^{n-k} (Z_{2} t)^{k} \\ &= \overline{t}^{n-k} t ^{k} Z_{1}^{n-k}Z_{2}^{k} \\ &\overset{}{=} t^{-n+2k} Z_{1}^{n-k}Z_{2}^{k} \\ \end{align}$which is precisely the $U_{n-2k}$ -action. Clearly the $V_{n}$ action is preserved on this $U_{n-2k}$ subspace. The result follows. - [ ] example on 2024 exam for character of SO(3) > [!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]: (Assuming finite) 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 $\lambda_{i} \equiv \rho_{g}$. - more properties to add (Sheet 4a) > [!basicproperties] > - $\chi_{\rho}(1)$ gives the dimension of the character $\chi$. > - $\chi_{\rho}$ is a [[class function]]: it is constant on [[conjugate|conjugacy classes]]. > - [[characters characterize complex group representations]]. > - $\chi_{\rho}(g^{-1})= \overline{\chi_{\rho}(g)}$. > - 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. (These were proved in the finite case) > **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 > ```