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zbjJFjJIeQ%(4oLp0_%rUZ*=*VO9T$)`MU&m9ls3)QARApQ_ydoap!}u#c1$vxe0^8 z)9?v{Hxn?U>Kd1bl77pQ2jd-Ag8q^35vmobfzJ$NCIU-`bcM$>ItthD zaJ2dX25C5VJ_J*!cb%7 diff --git a/skript.tex b/skript.tex index e29a619..802eeb3 100644 --- a/skript.tex +++ b/skript.tex @@ -1585,6 +1585,8 @@ $\set{S_1, \ldots, S_r}$ = vollständige Menge nicht isomorpher einfacher $A$-Mo \chapter{Charaktere} +\section{Charaktere} + $G$ = endliche Gruppe, alle $\C G$-Moduln sind endlich erzeugt. 6.1.1 Anmerkung: $G = $ endliche Gruppe $\Rightarrow^{\text{Maschke}} \C G$ ist semisimple.\\ @@ -1816,4 +1818,87 @@ Bemerkung: $\Hom_{\C G}(S_1^{\oplus \nu_1} \oplus \ldots \oplus S_r^{\oplus \nu_ 6.1.14 Korrolar: $\chi_1, \ldots, \chi_r$ ist eine Orthonormalbasis von $\Cl(G)$. Beweis: 6.1.13 impliziert, dass $(\chi_i, \chi_j) = \delta_{ij}$. \qed. +% paragraph 2 +\section{Charaktertafel} + +$G = $ Gruppe, $\abs{G} = n < \infty, \Irr(G) = \set{\chi_1, \ldots, \chi_r} = $ irreduzible Charakter von $G, \chi_1 $ trivialer Charakter. + +Fakten: +\begin{enumerate}[1)] + \item Jeder Charakter von $G$ ist $\Z$-Linearkombination der $\chi_i$ + \item Jeder Charakter von $G$ ist vollständig bestimmt durch seine Werte auf den Konjugationsklassen von $G$. + \item Die Anzahl der Konjugationsklassen von $G = r$. + \item Sind $C_1 \ni 1, \ldots, C_r$ die Konjugationsklassen von $G$, $g_i \in C_i$, so ist $\abs{C_i} = k_i = \abs{ G : C_G(g_i) }$ (1.3.13) + \item $\Irr(G)$ ist ONB vom $\C$-Vektorraum der Klassenfunktionen $\Cl(G)$ bezüglich des Skalarprodukts + $(\alpha, \beta) = \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha(g) \overline{\beta{g}} + = \frac{1}{\abs{G}} \sum\limits_{i=1}^r k_i \alpha(g_i) \overline{\beta(g_i)} = \sum\limits_{i=1}^r \frac{1}{\abs{C_G(g_i)}} \alpha(g_i) \overline{\beta(g_i)} $ +\end{enumerate} + +Definition: \underline{Charaktertafel} von $G$ ist eine $r \times r$-Matrix mit $\chi_i(g_j)$ als $ij$-Eintrag. \\ +Spaltenlabel: $g_i^{k_i}$, Zeilenlabel: $\chi_i$ (Atlas Notation) \\ + +$\chi_1(g_i) = 1, \chi_i(g_1) = f_i = $ Grad von $\chi_i$ = Dimension der zugehörigen Darstellungen += Größe des dazugehörigen Blocks von $\C G \cong \bigoplus\limits_{i=1}^r M_{f_i \times f_i}(\C)$ + +6.1.14 übersetzt sich in: \\ +6.2.1 Korrolar: $\delta_{ij} = \sum\limits_{t=1}^r \frac{1}{\abs{C_G(g_t)}} \chi_i(g_t) \overline{\chi_j(g_t)} $ \\ +Zeilen Der Charaktertafel sind orthonormal bezüglich es Standardskalarprodukt gewichtet mit $\frac{1}{\abs{C_G(g_t)}}$ auf $\C^r$. + +6.2.2 Korrolar: Seien $\alpha = \sum \lambda_i \chi_i, \beta = \sum \mu_j \chi_j$ virtuelle Charaktere. \\ +Dann ist $\alpha, \beta) = \sum\limits_{i=1}^r \lambda_i \mu_i$ + +6.2.3 Korrolar: Sei $\alpha$ virtueller Charakter (Klassenfunktion) von $G$. Dann ist $\alpha = \sum\limits_{i=1}^r (\alpha, \chi_i) \chi_i$ \\ +Insbesondere ist $\alpha$ Charakter von $G \Leftrightarrow (\alpha, \chi_i) \in \Z_{\geq 0}$ für $i = 1, \ldots, r$ und $\neq 0$ für mindestens ein $i$. + +Sofort: Ist $\chi$ ein Charakter von $G$, so ist $\chi$ irreduzibel $\Leftrightarrow (\chi, \chi) = 1$, also genau ein $\lambda_i = 1$ und der Rest $0$. + +6.2.4 Satz: Sei $\alpha$ linearer Charakter (i.e. $\alpha: G \rightarrow \C^\ast$ Homomorphismus) mit zugehörigem $\C G$-Modul $M$ ($M \cong \C$ als Vektorraum, $g \cdot m = \alpha(g) \cdot m$). \\ +Sei $X \in \lsub{\C G}{\underline{\Mod}}$ mit Charakter $\chi$. Dann ist $\alpha \chi$ ebenfalls Charakter von $G$ (mit Modul $M \otimes_\C X$), und $\alpha \chi$ ist irreduzibel! \\ +Beweis: $\alpha$ irreduzibel $\Rightarrow \alpha(g)$ Einheitswurzel $\forall g \in G$ da 1 = $\abs{\alpha(g)} = \alpha(g) \overline{\alpha(g)} \forall g \in G$. \\ +$(\alpha \chi, \alpha \chi) += \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha(g) \chi(g) \overline{\alpha(g) \chi(g)} += \frac{1}{\abs{G}} \sum\limits_{g \in G} \chi(g) \overline{\chi(g)} \alpha(g) \overline{\alpha(g)} += \frac{1}{\abs{G}} \sum\limits_{g \in G} \chi(g) \overline{\chi(g)} = (\chi, \chi) = 1 \Rightarrow $ Behauptung. + +Korrolar: Sei $\alpha$ Charakter von $G$, $(\alpha, \alpha) = n \in \set{1,2,3}$. Dann ist $\alpha$ Summe von $n$ irreduziblen Charakteren. + +6.2.5 Satz: (Spaltenorthogonalität): $\sum\limits_{t=1}^n \chi_t(g_i) \overline{\chi_t(g_i)} = \frac{\abs{G}}{k_i} \delta_{ij} = \abs{C_G(g_i)} \delta_{ij}$. \\ +Beweis: Sei $X = (\chi_i(g_j))_{ij}$ Charaktertafeln von $G$, $K = diag(k_1, \ldots, k_r) \in M_{r \times r}(\C)$. \\ +Dann ist $ij$-Eintrag von $X \cdot K = \chi_i(g_j) \cdot k_j$ und wir erhalten als $ij$-Eintrag von $X K X^t$ den Wert $\sum\limits_{l=1}^r \chi_i(g_l) k_l \overline{\chi_j(g_l)} += \sum\limits_{g \in G} \chi_i(g) \overline{\chi_j(g)} = \abs{G} (\chi_i, \chi_j) = \abs{G} \delta_{ij}$. \\ +Also ist $XK\overline{X}^t = \abs{G} E_{r \times r}$. \\ +Allgemein: $A, B \in M_{r \times r}(K), A \cdot B = \lambda E_{r \times r}, 0 \neq \lambda \in K \Rightarrow B = A^{-1} \lambda E = \lambda A^{-1}, BA = AB = \lambda E$ \\ +Also $K \overline{X}^t X = \abs{G} E_{r \times r}, (K \overline{X}^t X)_{ij} = \sum\limits_{l = 1}^r k_j \overline{chi_l(g_j)} \chi_l(g_i) = \abs{G} \delta_{ij}$ \qed. + +$K$ = Körper, $N \trianglelefteq G, \eta: G \to G / N: g \mapsto gN$ natürliche Projektion, $g_1, \ldots, g_s$ Nebenklassenvertreter von $N$ in $G$. \\ +Definiere $\hat{\eta}: KG \to K G/N: \sum\limits_{g \in G} \lambda_g g \mapsto \sum\limits_{g \in G} \lambda_g gN = \sum\limits_{i=1}^s (\sum\limits_{h \in g_i N} \lambda_h) g_i N$ \\ +Leichte Übung: $\hat{\eta}$ ist multiplikativ, d.h. ein $K$-Algebraepimorphismus ($\ker \hat{\eta} = < h - 1 \mid h \in N>_{\text{Ideal}}$) \\ +Damit kann jeder $K G/N$-Modul zu einem $KG$-Modul gemacht werden. + +Fast allgemein: Sei $A$ eine Algebra, $I \trianglelefteq A$ Ideal, $B = A/I$, $M \in \lsub{B}{\Mod}$. \\ +Dann wird $M$ zum $A$-Modul durch $a \cdot m := (a + I)m \forall a \in A, m \in M$. + +Richtig allgemein: $A, B$ seien $\Lambda$-Algebren, $f: A \to B$ Algebrahomomorphismus, $M \in \lsub{B}{\Mod}$. \\ +Dann wird $M$ zum $A$-Modul durch $a \cdot m := f(a) m \forall a \in A, m \in M$. \\ +Sind $M, N \lsub{B}{\Mod}, \beta: M \to N$ it $B$-linear, so ist $\beta$ auch $A$-linear. \\ +Also ist die Abbildung, die jedem $B$-Modul dem entsprechenden $A$-Modul zuordnet, ein Funktor, genannt "`Inflation"' entlang $f$. \\ +$ \Inf_{B, f}^A, A \to B \to \End_\Lambda(M) $ + +Sei $\chi$ Charakter von $G$, $K_\chi = \set{ x \in G \mid \chi(x) = \chi(1) }$ der Kern von $\chi$. (6.1.4) \\ +Dann ist $K_\chi \trianglelefteq G$. Für $\chi = \chi_i$ schreiben wir jetzt $K_i$ anstatt $K_{\chi_i}$. + +6.2.6 Satz: Sei $N \trianglelefteq G$, dann gibt es eine Teilmenge $I \subseteq \set{1, \ldots, r}$ so, dass $N = \bigcap\limits_{i \in I} K_i$ ist. \\ +Also: Die Normalteiler von $G$ sind extakt alle möglichen Schnitte der Kerne der irreduziblen Charakteren! \\ +Beweis: $U = \C (G/N), \psi = $ Charakter $\lsub{\C U}{U}$ und $\chi$ sei der entsprechende Charakter von $G$. ($g (h N) = (gh) N$) \\ +Sei $g \in G$: Dann ist $\chi(g) = \psi(gN) = \left\{\matr{\psi(1) = \abs{G/N} = \dim_\C U = \chi(1) & \text{für } g \in N \\ 0 & \text{sonst} }\right.$, d.h. $N = K_\chi$. \\ +($\abs{K_i} = \abs{ \bigcup\limits_{\chi_i(g_j) = \chi_i(1)}^{\cdot} g_j^G} = \sum\limits_{\chi_r(s_i) = \chi_r(1)} k_i$) +Sei $\chi = \sum\limits_{i=1}^r \alpha_i \chi_i$, 6.1.4 $\Rightarrow \abs{\chi(g)} \leq \sum a_i \abs{\chi_i(g)} \leq \sum \alpha_i \abs{\chi_i(1)} = \chi(1) \forall g \in G$. \\ +$ \sum \alpha_i \chi_i(g) = \chi(g) = \chi(1) = \sum \alpha_i \chi_i(1)$, also $\chi(g) = \chi(1) \Leftrightarrow (\chi_i(g) = \chi_i(1) $ oder $\alpha_i = 0) \forall i$ +Sei $I = \set{ 1 \leq i \leq r \mid \alpha_i \neq 0 }$ \\ +Also ist $N = \bigcap\limits_{i \in I} K_i$ + +6.2.7 Korrolar: $G$ einfach $\Leftrightarrow \forall 1 \neq g \in G \forall 2 \leq i \leq r: \chi_i(g) \neq \chi_i(1)$ + +6.2.8 Korrolar: Charaktertafel löst die Frage, ob $G$ auflösbar oder nicht auflösbar ist. + \end{document} diff --git a/standard.tex b/standard.tex index 5c7f067..68b1ae1 100644 --- a/standard.tex +++ b/standard.tex @@ -96,6 +96,7 @@ \DeclareMathOperator{\Irr}{Irr} \DeclareMathOperator{\tr}{tr} % Spur \DeclareMathOperator{\Cl}{Cl} +\DeclareMathOperator{\Inf}{Inf} % Inflation Funktor \DeclareMathOperator{\Reid}{Reid}