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+ +Definition: Sei $\varphi: \C G \rightarrow \End_{\C}(M)$ Darstellung, $\dim_{\C} M < \infty$. Dann heißt die Abbildung $\chi_M = \chi_{\varphi}: G \rightarrow \C: g \mapsto \tr(\varphi(g))$ ("`gewöhnlicher"') Charakter zur Darstellung $\varphi$. + +Beachte: $\varphi(g) \in \End_{\C}(M) \cong M_{n \times n}(\C)$ \\ +$\tr(m_f(\cB, \cB))$ ist unabhängig von der Wahl von $\cB$, da $\tr(AB) = \tr(BA)$, also $\tr(PAP^{-1}) = \tr(AP^{-1}P) = \tr(A)$. + +Also ist die Spur $\tr(f)$ für $f \in \End_{\C}(M)$ wohldefiniert, wenn wir $\tr(f) := \tr(m_f(\cB, \cB))$ setzen für irgendeine $\C$-Basis von $M$. + +Klar: Ist $M \cong N, M,N \in \lsub{\C G}{\Mod}$, so ist $\chi_M = \chi_N$. \\ +Seien $\varphi: G \rightarrow \Aut_\C(M) \rightarrow \GL_m(\C) (\dim_\C M = m)$ \\ +und $\psi: G \rightarrow \Aut_\C(N) \rightarrow \GL_n(\C) (\dim_\C N = n)$ \\ +Erinnerung: Sei $\tau: M \rightarrow N$ $\C G$-Isomorphismus. Dann gilt: $m = n$ und $\forall g \in G$: \\ +\begin{diagram} +M & \rTo^{\varphi(g)} & M & ~~ & \C^n & \rTo^{\tilde{\varphi}(g)} & \C^n \\ +\dTo^{\tau} & & \dTo^{\tau} & ~~ & \dTo^{\tau} & & \dTo^{\tau} \\ +N & \rTo^{\psi(g)} & N & ~~ & \C^n & \rTo^{\tilde{\psi}(g)} & \C^n +\end{diagram} +$\tau \tilde{\varphi}(g) = \tilde{\psi}(g) \tau \Leftrightarrow \tilde{\varphi}(g) = \tau^{-1} \tilde{\psi}(g) \tau$ \\ +$\Rightarrow \chi_M(g) = \tr(\tilde{\varphi}(g)) = \tr(\tilde{\psi}(g)) \forall g \in G$ + +Beispiel: +\begin{enumerate}[i)] + \item Sei $M = \C_G$ triviale $\C G$-Modul, d.h. $(\sum \lambda_g g) z = (\sum \lambda_g) z, ~~ (\lambda_g, z \in \C)$ \\ + $\chi_M: g \rightarrow 1 \in \C$ + \item Sei $\Omega$ eine transitive $G$-Menge, $M_\Omega =$ zugehöriger $\C G$-Modul. \\ + Ohne Einschränkung $\Omega = \set{1, \ldots, n}$, $G \rInto \sigma_n \cong W \leq GL_n(\C)$ \\ + Sei $P_\pi = $ Permutationsmaatrix zu $\pi \in \sigma_n, \tr(P_\pi) = \abs{\set{i \in \set{1, \ldots, n} \mid \pi(i) = i}}$. \\ + $\tr(g) = \abs{\set{w \in \Omega \mid gw = w}}$ + +Spezialfall: $M = \lsub{\C G}{\C G}$. \\ +Sei $\chi_M$ zugehöriger Charakter. \\ +$\chi_M(g) = \abs{\set{x \in G \mid gx = x}} = \left\{\matr{0 & \text{für } g \neq 1 \\ \abs{G} & \text{für } g = 1}\right.$ + + \item Sei $\chi: G \rightarrow \C$ Charakter zum $\C G$-Modul $M$. \\ + $\chi(1) = \dim_{\C} M$. +\end{enumerate} + +Bezeichnung: Seien $S_1, S_2, \ldots, S_r$ die verschiedenen irreduziblen $\C G$-Moduln, wobei $S_1$ der triviale $\C G$-Modul ist. Wir bezeichnen +den Charakter $\chi_{S_i}$ des $i$-ten $\C G$-Modules mit $\chi_i (i = 1, \ldots, r)$ \\ +$\Irr(G) = \set{\chi_1, \ldots, \chi_r}$. \\ +Die $\chi_i$ heißen irreduzible Charaktere. \\ +Sei $f_i = \dim_\C S_i: \C G = \bigoplus\limits_{i=1}^r M_{f_i \times f_i}(\C)$ \\ + +Bemerkung: Ist $\dim_\C M = 1$ (z.Bsp. $S_1$), so spricht man von einem linearen Charakter, denn $\chi_M : G \rightarrow \C^\ast$ ist Gruppenhomomorphismus. + +6.1.4 Satz: Sei $U \in \lsub{\C G}{\Mod}, \rho: G \rightarrow \GL_\C(U) \subseteq \End_\C(U)$ zugehörige Darstellung. Sei $g \in G, \abs{g} = n$. \\ +Dann gilt: +\begin{enumerate}[i)] + \item $\rho(g) \in \End_{\C}(U)$ ist diagonalisierbar + \item $\chi_U(g) = $ Summe (mit Vielfachheiten) der Eigenwerte von $\rho(g)$ (LAAG 2: Jordansche NF) + \item $\chi_U(g) = $ Summe von $\chi_U(1) = \dim_\C U$ vielen $n$-ten Einheitswurzeln. + \item $\chi_U(g^{-1}) = \overline{\chi_U(g)}$ + \item $\abs{\chi_U(g)} = \sqrt{\chi_U(g) \overline{\chi_U(g)}} \leq \chi_U(1)$ + \item $\set{g \in G \mid \chi_U(x) = \chi_U(1) = \dim_\C U} \trianglelefteq G$. +\end{enumerate} + +Beweis: +\begin{enumerate}[i)] + \item $g^n = 1 \Rightarrow (\rho(g))^n = \id_U \rightsquigarrow E_{\dim_\C(U)} \Rightarrow \rho(g)$ genügt dem Polynom $x^n-1 \in \C[X] $ + $\Rightarrow $ Minimalpolynom $\mu_{\rho(g)}(t)$ hat nur einfache Nullstellen $\Rightarrow \rho(g)$ ist diagonalisierbar. + \item Folgt aus i) + \item Folgt aus i) + \item Jeder Eigenvektor von $\rho(g)$ zu einem Eigenwert $z \in \C$ ist Eigenvektor von $\rho(g^{-1})$ zum Eigenwert $z^{-1} \in \C:$ \\ + $\rho(g) v = z v \Rightarrow z^{-1} v = (\rho(g)^{-1})(v) = \rho(g^{-1})(v), z\overline{z} = \abs{z}^2 = 1^2 = 1 \Rightarrow \overline{z} = z^{-1}$ \\ + $\chi_{\rho}(g) = \sum $ Einheitswurzeln, $\chi_{\rho^{-1}} = \sum \overline{\text{Einheitswurzeln}} = \overline{ \sum \text{Einheitswurzeln}} = \overline{\chi_U(g)}$ \\ + $ \abs{\chi_U(g)} = \abs{ \sum\limits_{\dim_\C U} \text{Eigenwerte } h} \mathop{\leq}\limits^{\text{Dreiecks-Ugl.}} \sum\limits_{\dim_\C(U)} \abs{\text{Eigenwert h}} = \dim_\C U = \chi_U(1)$. + \item[vi)] $\abs{z_1 + z_2} = \abs{z_1} + \abs{z_2} \Leftrightarrow z_1 = \lambda z_2 \exists \lambda \in \R^{+}$ + \item[v)] $ \Rightarrow \abs{\chi_U(g)} = \chi_U(1) \Leftrightarrow$ ale Eigenwerte von $\rho(g)$ sind 1 $\Leftrightarrow \rho(g) = \id_U \Leftrightarrow g \in \ker \rho_U \trianglelefteq G$. \\ + $ \rho_U: G \Aut_\C(U)$ +\end{enumerate} + +Erinnerung: $\C^G = \set{ \alpha : G \rightarrow \C }$ ist $\C$-Algebra mit Addition $(\alpha_1 + \alpha_g)(g) = \alpha_1(g) + \alpha_2(g), (\alpha_1 \cdot \alpha_2)(g) = \alpha_1(g) \cdot \alpha_2(g)$ und +$(\lambda \alpha)(g) = \lambda (\alpha(g))$. \\ +Menge Der Charaktere von $G \subseteq \C^G$. + +6.1.5 Lemma: $(\chi_1, \ldots, \chi_r)$ ist linear unabhängig in dem $\C$-Vektorraum $\C^G$. \\ +Beweis: $\C G \cong M_{f_1 \times f_1}(\C) \oplus \ldots \oplus M_{f_r \times f_r}(\C)$, $S_i=$ der eindeutig bestimmte irreduzible $G$-Modul im Block $M_{f_i \times f_i}(\C)$. \\ +Sei $e_i = $ Einselement von $M_{f_i \times f_i}(\C)$ (also $e_i = ( (0)_{f_1 \times f_1}, \ldots, (0)_{f_{i-1} \times f_{i-1}}, (1)_{f_i \times f_i}, (0) \ldots)$) Idemp. von $\C G$. \\ +Seien $\chi_1, \ldots, \chi_r$ die zu $S_i$ gehörenden Charaktere von $G$. \\ +Wir können $\chi_i$ linear zu einer lineare Abbildung vn $\C G$ nach $\C$ ausdehnen: $\chi_i \in \Hom_\C(\C G, \C) = \C G^\ast$, d.h.für $a = \sum\limits_{g\in G} \alpha_g g \in \C G$ +ist $\chi_i(\alpha) = \sum\limits_{g \in G} \alpha_g \chi_i(g) \in \C$. \\ +Beachte: Auf $S_i$ operiert $e_i$ wie die Eins. D.h. $\chi_i(e_i) = \dim_\C(S_i) = f_i$ und $\chi_j(e_i) = 0$ für $i \neq j$. \\ +Seien $\lambda_1, \ldots, \lambda_r \in \C$ so, dass $\sum\limits_{j=1}^r \lambda_j \chi_j = 0$ ist. Dann ist für $1 \leq i \leq r: 0 = \sum\limits_{j=1}^r \lambda_j \chi_j)(e_i) = \lambda_i \chi_i(e_i) \Rightarrow \lambda_i = 0 \forall i$. Also sind $(\chi_1, \ldots, \chi_r)$ linear unabhängig. + +6.1.6 Lemma: $U, V \in \lsub{\C G}{\Mod}$ endlich erzeugt. Dann $\chi_{U \oplus V} = \chi_U + \chi_V$ \\ +Beweis: Trivial. ($\rho_{U \oplus V}(g)$ hat Blockdiagonalform. Spur nehmen $\Rightarrow$ Behauptung.) + +Beachte: Sind $g, h \in G$ konjugiert, $\chi$ Charakter von $G \Rightarrow \chi(g) = \chi(u)$, +denn $g, h$ konjugiert $\Rightarrow \exists x \in G: g = xhx^{-1} \Rightarrow \chi(g) = \chi(xhx^{-1}) = \chi((hx^{-1})x) = \chi(h)$, +d.h. $\chi$ ist konstant auf Konjugationsklassen von $G$. \\ +Abbildung $\alpha: G \rightarrow \C (\alpha \in \C^G)$ heißen \underline{Klassenfunktionen}, wenn $\alpha(g) = \alpha(h)$ falls $g, h$ konjugiert sind. + +Wir haben gezeigt: $\chi_1, \ldots, \chi_r$ sind $r$ viele linear unabhängig Klassenfunktionen in $\C^G$. \\ +Definiere: $C_1, \ldots, C_r$ seien die Konjugationsklassen auf $G$. Für $1 \leq i \leq r$ sei $\epsilon_i: g \mapsto \left\{\matr{1 & \text{falls } g \in C_i\\0 & \text{sonst}}\right.$ \\ +Klar: $\set{\epsilon_1,\ldots, \epsilon_r}$ linear unabhängig in der Unteralgebra der Klassenfunktionen. \\ +Sei $\alpha: G \rightarrow \C$ Klassenfunktion, mit $\alpha(g) = \alpha_i \in \C$ für $g \in C_i \Rightarrow \alpha = \alpha_1 \epsilon_i + \ldots + \alpha_r \epsilon_r$ + +Folgerung: $\set{\chi_1, \ldots, \chi_r}$ ist Basis der $\C$-Algebra der Klassenfunktionen auf $G$. + +Bemerkung: Sei $M \in \lsub{\C G}{\Mod}$ endlich erzeugt $\Rightarrow \exists \nu_1, \ldots, \nu_r \in \N_0$ so, dass $M = S_1^{\oplus \nu_1} \oplus \ldots \oplus S_r^{\oplus \nu_r}$ +$\Rightarrow \chi_M = \nu_1 \chi_1 + \ldots + \nu_r \chi_r$. + +Es gilt: Ist $M \ncong N$ und $M, N \in \lsub{\C G}{\Mod}$ endlich erzeugt $\Rightarrow \chi_M \neq \chi_N$. + \end{document} diff --git a/standard.tex b/standard.tex index d9f1b14..4023dd4 100644 --- a/standard.tex +++ b/standard.tex @@ -93,6 +93,8 @@ \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Mod}{mod} \DeclareMathOperator{\Ind}{Ind} +\DeclareMathOperator{\Irr}{Irr} +\DeclareMathOperator{\tr}{tr} % Spur \DeclareMathOperator{\Reid}{Reid} \DeclareMathOperator{\Liid}{Liid}

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