OrthogonalityRelations 2003-01-05 17:08:18 2007-08-03 20:32:47 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{thm}{Theorem} \newtheorem{prop}{Proposition} \newcommand{\ab}[1]{{#1}_{\mathrm{ab}}} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\Aut}{\mathrm{Aut}\,} \newcommand{\Aff}[2]{\mathrm{Aff}_{#1} #2} \newcommand{\aff}[2]{\mathfrak{aff}_{#1} #2} \newcommand{\mcB}{\mathcal{B}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\bfrac}[2]{\left[\frac{#1}{#2}\\right]} \newcommand{\bkh}{\backslash} \newcommand{\Cyc}[2]{\mathcal{C}^{#1}_{#2}} \newcommand{\Cbar}[2]{\overline{\C{#1}{#2}}} %\newcommand{\CD}{\R[\Delta]} \newcommand{\C}{\mathbb{C}} \newcommand{\CF}[2]{\ensuremath{\mathfrak{C}(#1,#2)}} \newcommand{\Cinf}{\EuScript{C}^{\infty}} \newcommand{\cmp}{cyclic mod $p$\xspace} \newcommand{\cp}{\mathrm{c.p.}} \newcommand{\CS}{\EuScript{CS}} \newcommand{\deck}{\EuScript{D}} \newcommand{\defl}[1]{\mathfrak{def}_{#1}} \newcommand{\Der}{\mathrm{Der}\,} \newcommand{\eH}{[X_H]-[Y_H]} \newcommand{\EL}{\mathcal{EL}} \newcommand{\End}{\mathrm{End}} \newcommand{\ES}[1]{\EuScript{#1}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Fix}{\mathrm{Fix}} \newcommand{\fr}[1]{\mathfrak{#1}} \newcommand{\Frat}{\mathrm{Frat}\,} \newcommand{\Gal}[1]{\Gamma(#1 |\Q)} \newcommand{\GL}[2]{\mathrm{GL}_{#1} #2} \newcommand{\gl}[2]{\mathfrak{gl}_{#1} #2} \newcommand{\GrR}[1]{a(#1 G)} \newcommand{\Gr}{\mathrm{Gr}\,} \newcommand{\mcH}{\mathcal{H}} \renewcommand{\H}{\mathbb{H}} \newcommand{\Hom}[2]{\mathrm{Hom}(#1,#2)} \newcommand{\id}{\mathrm{id}} \newcommand{\im}{\mathrm{im}} \newcommand{\ind}[2]{\mathrm{ind}^{#1}_{#2}} \newcommand{\indp}[2]{\mathfrak{ind}^{#1}_{#2}} \renewcommand{\inf}[1]{\mathfrak{inf}_{#1}} \newcommand{\inn}[1]{\langle #1\rangle} \renewcommand{\int}{\mathrm{int}} \newcommand{\Iso}{\mathrm{Iso}} \newcommand{\K}{\mathcal{K}} \renewcommand{\ker}{\mathrm{ker}\,} \renewcommand{\L}[1]{\mathfrak{L}(#1)} \newcommand{\lap}[1]{\Delta_{#1}} \newcommand{\lapM}{\Delta_M} \newcommand{\Lie}{\mathrm{Lie}} \newcommand{\lineq}{linearly equivalent\xspace} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mG}{m_G} \newcommand{\mK}{m_{\K}} \newcommand{\mindeg}[1]{\fr{md}(#1)} \newcommand{\N}{\mathbb{N}} \renewcommand{\O}{\mathcal{O}} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\Orb}{\mathrm{Orb}} \newcommand{\pad}{\hat{\Z}_p} \newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pderw}[1]{\frac{\partial}{\partial #1}} \newcommand{\pdersec}[2]{\frac{\partial^2 #1}{\partial {#2}^2}} \newcommand{\perm}[1]{\pi_{#1}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\rad}{\mathrm{rad}\,} \newcommand{\res}[2]{\mathrm{res}^{#1}_{#2}} \newcommand{\resp}[2]{\mathfrak{res}^{#1}_{#2}} \newcommand{\RG}{\EuScript{R}_G} \newcommand{\rk}{\mathrm{rk}\,} \newcommand{\V}[1]{\mathbf{#1}} \newcommand{\vp}{\varphi} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\SL}[2]{\mathrm{SL}_{#1} #2} \renewcommand{\sl}[2]{\fr{sl}_{#1} #2} \newcommand{\SO}[2]{\mathrm{SO}_{#1} #2} \newcommand{\Sp}[2]{\mathrm{Sp}_{#1} #2} \renewcommand{\sp}[2]{\fr{sp}_{#1} #2} \newcommand{\SU}[1]{\mathrm{SU}( #1)} \newcommand{\su}[1]{\fr{su}_{#1}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\sym}{\mathrm{sym}} \newcommand{\Tg}{\mc{T}(\fr g)} \newcommand{\tom}{\tilde{\omega}} \newcommand{\ghtghp}{\fr g/\fr h\oplus(\fr g/\fr h^\perp)^*} \newcommand{\ghps}{(\fr g/\fr h^\perp)^*} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\tr}{\mathrm{tr}} %\renewcommand{\thechapter}{\Roman{chapter}} %\renewcommand{\thesection}{\thechapter.\arabic{section}} %\renewcommand{\thethm}{\thechapter.\arabic{thm}} \newcommand{\Ug}{\mc{U}(\fr g)} \newcommand{\Uh}{\mc{U}(\fr h)} \renewcommand{\V}[1]{\mathbf{#1}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Zp}{\Z/p} \textbf{First orthogonality relations}: Let $\rho_\alpha\colon G \to V_\alpha$ and $\rho_\beta\colon G \to V_\beta$ be irreducible representations of a finite group $G$ over the field $\C$. Then \[ \frac{1}{|G|}\sum_{g\in G}\overline{\rho^{(\alpha)}_{ij}(g)}\rho^{(\beta)}_{kl}(g)=\frac{\delta_{\alpha\beta}\delta_{ik}\delta_{jl}}{\dim V_\alpha}. \] We have the following useful corollary. Let $\chi_1$, $\chi_2$ be characters of representations $V_1$, $V_2$ of a finite group $G$ over a field $k$ of characteristic $0$. Then $$(\chi_1,\chi_2)=\frac{1}{|G|}\sum_{g\in G}\overline{\chi_1(g)}\chi_2(g)=\dim(\Hom{V_1}{V_2}).$$ \begin{proof} First of all, consider the special case where $V=k$ with the trivial action of the group. Then $\mathrm{Hom}_G(k,V_2)\cong V_2^G$, the fixed points. On the other hand, consider the map $$\phi=\frac{1}{|G|}\sum_{g\in G} g\colon V_2\to V_2$$ (with the sum in $\mathrm{End}(V_2)$). Clearly, the image of this map is contained in $V_2^G$, and it is the identity restricted to $V_2^G$. Thus, it is a projection with image $V_2^G$. Now, the rank of a projection (over a field of characteristic 0) is its trace. Thus, $$\dim_k \mathrm{Hom}_G(k,V_2)=\dim V_2^G=\mathrm{tr}(\phi)= \frac{1}{|G|}\sum\chi_2(g)$$ which is exactly the orthogonality formula for $V_1=k$. Now, in general, $\mathrm{Hom}(V_1,V_2)\cong V_1^*\otimes V_2$ is a representation, and $\mathrm{Hom}_G(V_1,v_2)=(\mathrm{Hom}(V_1,V_2))^G$. Since $\chi_{V^*_1\otimes V_2}=\overline{\chi_1}\chi_2$, $$\dim_k\mathrm{Hom}_G(V_1,V_2)=\dim_k (\mathrm{Hom}(V_1,V_2))^G= \sum_{g\in G}\overline{\chi_1}\chi_2$$ which is exactly the relation we desired. \end{proof} In particular, if $V_1, V_2$ irreducible, by Schur's Lemma $$\Hom{V_1}{V_2}=\begin{cases} D &V_1\cong V_2\\ 0&V_1\ncong V_2\end{cases}$$ where $D$ is a division algebra. In particular, non-isomorphic irreducible representations have orthogonal characters. Thus, for any representation $V$, the multiplicities $n_i$ in the unique decomposition of $V$ into the \PMlinkname{direct sum}{DirectSum} of irreducibles $$V\cong V_1^{\oplus n_1}\oplus\cdots\oplus V_m^{\oplus n_m}$$ where $V_i$ ranges over irreducible representations of $G$ over $k$, can be determined in terms of the character inner product: $$n_i=\frac{(\psi,\chi_i)}{(\chi_i,\chi_i)}$$ where $\psi$ is the character of $V$ and $\chi_i$ the character of $V_i$. In particular, representations over a field of characteristic zero are determined by their character. Note: This is not true over fields of positive characteristic. If the field $k$ is algebraically closed, the only finite division algebra over $k$ is $k$ itself, so the characters of irreducible representations form an orthonormal basis for the vector space of class functions with respect to this inner product. Since $(\chi_i,\chi_i)=1$ for all irreducibles, the multiplicity formula above reduces to $n_i=(\psi,\chi_i)$. \textbf{Second orthogonality relations}: We assume now that $k$ is algebraically closed. Let $g,g'$ be elements of a finite group $G$. Then $$\sum_{\chi}\chi(g)\overline{\chi(g')}=\begin{cases}|C_G(g_1)|&g\sim g'\\ 0&g\nsim g'\end{cases}$$ where the sum is over the characters of irreducible representations, and $C_G(g)$ is the centralizer of $g$. \begin{proof} Let $\chi_1,\ldots,\chi_n$ be the characters of the irreducible representations, and let $g_1,\ldots,g_n$ be representatives of the conjugacy classes. Let $A$ be the matrix whose $ij$th entry is $\sqrt{|G:C_G(g_j)|}(\overline{\chi_i(g_j)})$. By first orthogonality, $AA^{*}=|G|I$ (here $*$ denotes conjugate transpose), where $I$ is the identity matrix. Since left \PMlinkname{inverses}{MatrixInverse} are right \PMlinkescapetext{inverses}, $A^{*}A=|G|I$. Thus, $$\sqrt{|G:C_G(g_i)||G:C_G(g_k)|}\sum_{j=1}^n\chi_j(g_i)\overline{\chi_j(g_k)}=|G|\delta_{ik}.$$ Replacing $g_i$ or $g_k$ with any conjuagate will not change the expression above. thus, if our two elements are not conjugate, we obtain that $\sum_\chi\chi(g)\overline{\chi(g')}=0$. On the other hand, if $g\sim g'$, then $i=k$ in the sum above, which reduced to the expression we desired. \end{proof} A special case of this result, applied to $1$ is that $|G|=\sum_{\chi}\chi(1)^2$, that is, the sum of the squares of the \PMlinkname{dimensions}{Dimension} of the irreducible representations of any finite group is the order of the group.