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Es gilt: Ist $M \ncong N$ und $M, N \in \lsub{\C G}{\Mod}$ endlich erzeugt $\Rightarrow \chi_M \neq \chi_N$. +6.1.7 Satz: Zwei endliche erzeugte $\C G$-Moduln sind isomorph $\Leftrightarrow$ ihre Charakter stimmen überein. \\ +Beweis: Sind $S_1, \ldots, S_r$ die verschiedenen irreduziblen $\C G$-Moduln mit Charakteren $\chi_1, \ldots, \chi_r, M, N \in \lsub{\C G}{\Mod}$. +Sei $M = S_1^{\oplus \mu_1} \oplus \ldots \oplus S_r^{\oplus \mu_r}, N = S_1^{\oplus \nu_1} \oplus \ldots \oplus S_r^{\oplus \nu_r}$, so ist +$\chi_M = \sum\limits_{i=1}^r \mu_i \chi_i, \chi_N = \sum\limits_{i=1}^r \nu_i \chi_i$ \\ +$M \cong N \Leftrightarrow \mu_i = \nu_i \forall i = 1, \ldots, r \Leftrightarrow \chi_M = \chi_N$ \qed + +Bemerkung: Es gibt keinen kanonischen Weg, aus dem Charakter eines $\C G$-Moduls $M$ den Modul $M$ selbst zu bestimmen. + +Erinnerung: $U, V \in \lsub{\C G}{\Mod} \Rightarrow U \otimes_\C V, \Hom_\C(U, V), U^\ast = \Hom_\C(U, \C)$ sind $\C G$-Moduln. + +6.1.8 Satz: $U, V \in \lsub{\C G}{\Mod}$. Dann gilt: +\begin{enumerate}[i)] + \item $\chi_{U \times V}= \chi_U \cdot \chi_V (\chi_{U \otimes V}(g) = \chi_U(g) \cdot \chi_V(g))$ + \item $\chi_{U^\ast} = \overline{\chi_U} (\chi_{U^\ast}(g) = \overline{\chi_U(g)}$ + \item $\chi_{\Hom_\C(U, V)} = \overline{\chi_U} \cdot \chi_V$ +\end{enumerate} +Beweis: +\begin{enumerate}[i)] + \item Sei $g \in G$. Sei $(u_1, \ldots, u_m)$ und $(v_1, \ldots, v_n)$ Basen von $U$ bzw. $V$ aus Eigenvektoren von $g$ auf $U$ bzw. $V$. + D.h. $\exists \mu_1, \ldots, \mu_m, \nu_1, \ldots, \nu_n \in \C: g u_i = \mu_i u_i$ und $g v_j = \nu_j v_j \forall 1 \leq i \leq m, 1 \leq j \leq n$. \\ + Dies geht nach 6.1.4, da die Operation von $G$ auf $U$ bzw. $V$ diagonalisierbar ist. \\ + LAAG: Die $( u_i \otimes v_j \mid 1 \leq i \leq m, 1 \leq j \leq n )$ ist Basis von $U \otimes V$ \\ + $g(u_i \otimes v_i) = (g u_i) \otimes (g v_j) = (\mu_i u_i) \otimes (\nu_j v_j) = \mu_i \nu_j (u_i \otimes v_j)$ \\ + Es folgt: $\chi_{U \otimes V}(g) = \sum\limits_{i,j} \mu_i \nu_j = (\sum\limits_{i=1}^m \mu_i) (\sum\limits_{j=1}^n \nu_j) = \chi_U(g) \chi_V(g)$ + \item $g \in G, (u_1, \ldots, u_m)$ Basis aus Eigenvektoren von $g$ auf $U$, $g u_i = \mu_i u_i, \mu_i \in \C, 1\leq i \leq m$. \\ + Sei $(u_1^\ast, \ldots, u_m^\ast)$ Basis von $U^\ast$, d.h. $u_i^\ast(u_j) = \delta_{ij}$. + $\Rightarrow (g u_i^\ast)(u_j) = u_i^\ast(g^{-1} u_j) \mathop{=}\limits^\text{6.1.4} u_i^\ast (\mu_j u_j) = \overline{\mu_j} \delta_{ij}$ \\ + $\Rightarrow g u_i^\ast = \overline{\mu_i} u_i^\ast$, d.h. $u_i^\ast$ ist Eigenvektor von Operation von $g$ auf $U^\ast$ mit Eigenwerten $\overline{\mu_i}$. \\ + Also ist $\chi_{U^\ast}(g) = \sum\limits_{i=1}^m \overline{\mu_i} = \overline{\sum\limits_{i=1}^m \mu_i} = \overline{\chi_U(g)}$. \\ + \item Nach 5.3.4 (?) ist $\Hom_\C(U, V) \cong U^\ast \otimes V$ als $\C G$-Modul. Die Behauptung folgt aus i) und ii). \qed +\end{enumerate} + +Bemerkung: $\set{\Char \text{ von } G} \subseteq \C^G = \C$-Algebra und darin sind die irreduziblen Charaketere $(\chi_1, \ldots, \chi_r)$ linear unabhängig. +So ist $<\chi_1, \ldots, \chi_r>_{\C\text{-Aufspann}}$ $r$-dimensionaler Unterraum von $\C^G$. \\ +Abelsche Gruppe, freier $\Z$-Modul: $\set{ \sum\limits_{i=1}^r \mu_i \chi_i \mid \mu_i \in \Z} \subseteq <\chi_1, \ldots, \chi_r>$. \\ +Klassenfunktionen der Form $\sum\limits_{i=1}^r \mu_i \chi_i$ mit $\mu_i \in \Z$ heißen "`virtuelle"' Charaktere. + +6.1.9 Korrolar: Der Raum der virtuellen Charaktere ist ein Ring. \\ +Beweis: Nach 6.1.8 ist das Produkt zweier Charaktere wieder ein Charakter, d.h. ganzzahlige Linearkombination der $\chi_1, \ldots, \chi_r$. Also ist auch das Produkt +zweier virtueller Charaktere wieder ein virtueller Charakter. \qed + +Definition: Eine Klassenfunktion auf $G$ ist eine Abbildung $f: G \rightarrow \C$ mit $f(g) = f(h)$ falls $g, h$ konjugiert (also $g \sim_G h$). \\ +Klar: Summe, lineare Vielfache und Produkte vo Klassenfunktionen sind Klassenfunktionen. \\ +Also: $< \chi_1, \ldots, \chi_r >_\C \subseteq \set{\text{Klassenfunktionen auf } G} \leq \C^G$ + +6.1.10 $<\chi_1, \ldots, \chi_r>_\C$ ist die $\C$-Algebra der Klassenfunktionen.auf $G$. \\ +Beweis: Seien $C_1, \ldots, C_r$ die Konjugationsklassen von $G$, $1 \leq i, j \leq r$. Sei $\psi_i: G \rightarrow \C$ definiert durch: \\ +$\psi_i(g) = \left\{\matr{1 & \text{für } g \in C_i \\ 0 & \text{sonst}}\right.$ \\ +Sei $f: G \rightarrow \C$ Klassenfunktion mit $f(g) = \lambda_i$ für $g \in C_i$ +$ \Rightarrow f = \sum\limits_{i=1}^r \lambda_i \psi_i \Rightarrow f \in < \psi_1, \ldots, \psi_r>_\C$ \\ +Trivial: $\psi_1, \ldots, \psi_r$ linear unabhängig $ \Rightarrow \dim_\C \set{\text{Klassenfunktionen auf } \C} = r$ \\ +$\Rightarrow $ Behauptung. \qed + +Definition: $\Cl(G) := \C$-Vektorraum der Klassenfunktionen auf $G$. \\ +Auf $\Cl(G)$ definieren wir ein inneres Produkt $( \cdot, \cdot )$ durch: $(\alpha, \beta \in \Cl(G))$ \\ +$(\alpha, \beta) = \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha(g) \overline{\beta(g)} \in \C$ \\ +Wir haben: \\ +$ ~~~~ (\alpha, \alpha) = \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha(g) \overline{\alpha(g)} = \frac{1}{\abs{G}} \sum\limits_{g \in G} \abs{\alpha(g)}^2 = 0 +\Leftrightarrow \alpha(g) = 0 \forall g \in G \Leftrightarrow \alpha = 0$. \\ +$ ~~~~ (\alpha, \beta) = \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha(g) \overline{\beta(g)} = \frac{1}{\abs{G}} \sum\limits_{g \in G} \overline{\beta(g) \overline{\alpha(g)}} = \overline{(\beta, \alpha)}$ \\ +$ ~~~~ (\lambda \alpha, \beta) = \frac{1}{\abs{G}} \sum\limits_{g \in G} (\lambda \alpha)(g) \overline{\beta(g)} = \frac{\lambda}{\abs{G}} \sum\limits_{g \in G} \alpha(g) \overline{\beta(g)} = \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha(g) \overline{(\overline{\lambda}\beta)(g)} = (\alpha, \overline{\lambda} \beta)$ \\ +$ ~~~~ (\alpha_1 + \alpha_2, \beta) +% = \frac{1}{\abs{G}} \sum\limits_{g \in G} (\alpha_1 + \alpha_2)(g) \overline{\beta(g)} + = \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha_1(g) \overline{\beta(g)} + \frac{1}{\abs{G}} \sum\limits_{g \in G} \alpha_2(g) \overline{\beta(g)} = (\alpha_1, \beta) + (\alpha_2, \beta)$. + +Also ist $(\cdot, \cdot)$ wirklich ein Skalarprodukt auf $\Cl()G)$. + +Erinnerung: $U \in \lsub{\C G}{\underline{\Mod}}$ endlich erzeugt. Dann ist $U^G = \set{u \in U \mid gu = u \forall g \in G}$ ist der eindeutig +bestinmmte größte Untermodul von $U$, auf dem $G$ trivial operiert. \\ +$ U = S_1^{\oplus \mu_1} \oplus \ldots \oplus S_r^{\oplus \mu_r}, S_1 = $ trivialer $\C G$-Modul. \\ +$\Rightarrow U^G = S_1^{\oplus \mu_1}, \mu_1 = \dim_\C U^G$ + +Beobachtung: $U^G = $ simultae Eigenraum $\forall g \in G$ zum Eigenwert 1 = $\bigcap\limits_{g \in G}$ Eigenraum von $g$ zum Eigenwert $1$. + +6.1.11 Lemma: Sei $U \in \lsub{\C G}{\underline{\Mod}}$ endlich erzeugt, dann ist $\dim_\C U^G = \frac{1}{\abs{G}} \sum\limits_{g \in G} \chi_U(g) ( = \chi_U(\frac{1}{\abs{G}} \sum\limits_{g \in G} g = \chi_U(T))$. \\ +Beweis: Sei $a = \frac{1}{\abs{G}} \sum\limits_{g \in G} g \in \C G$. Sei $h \in G \Rightarrow ha = \frac{1}{\abs{G}} \sum\limits_{g \in G} h g = \frac{1}{\abs{G}} \sum\limits_{g \in G} g = a = ah$ \\ +$\Rightarrow a^2 = (\frac{1}{\abs{G}} \sum\limits_{g \in G} g) a = \frac{1}{\abs{G}} \sum\limits_{g \in G} (g a) = \frac{1}{\abs{G}} \sum\limits_{g \in G} a = a, a $ ist Idempotent von $\C G$. \\ +$(\C G = M_{1 \times 1}(\C) \oplus M_{f_2 \times f_2}(\C) \oplus \ldots \oplus M_{f_r \times f_r}(\C); a = ( 1, 0, \ldots, 0 )$. \\ +Sei $\rho: \C G \rightarrow \End_\C(U)$ die zu $U \in \lsub{\C G}{\underline{\Mod}}$ gehörende Darstellung von $\lsup{\C}G$ und sei $T = \rho(a)$. \\ +Dann gilt $T^2 = (\rho(a))^2 = \rho(a^2) = \rho(a) = T$, d.h. $T$ erfüllt das Polynom $X^2 - X \in \C[X] \Rightarrow 0$ und $1$ sind die einzigen möglichen Eigenwerte von $T$ und $T$ ist diagonalisierbar. + +Sei $U_1 \subseteq U$ der Eigenraum von $T$ zum Eigenwert $1$. \\ +$u \in U_1 \Rightarrow g(u) = g(a \cdot u) = (g a)(u) = a u = u$ \\ +$\Rightarrow U_1 \leq U^G$ \\ +Sei $u \in U^G$, dann ist $\abs{G} a u = (\sum\limits_{g \in G} g) u = \sum\limits_{g \in G} (gu) = \sum\limits_{g \in G} u = \abs{G} u \Rightarrow au = u \Rightarrow u \in U_1$ \\ +Also ist $U_1 = U^G$. Die Spur $\tr(T)$ von $T$ auf $U$ ist $\dim_\C(U_1)$ + +6.1.12 Erinnerung: Sind $U, V \in \lsub{\C G}{\underline{\Mod}}$ endlich erzeugt. Dann ist $\Hom_\C(U, V) = U^\ast \otimes V \in \lsub{\C G}{\underline{\Mod}}$ und es +ist $\Hom_{\C G}(U, V) = (\Hom_\C(U, v))^G$. + +6.1.13 Korrolar: Seien $U, v \in \lsub{\C G}{\underline{\Mod}}$. Dann ist $(\chi_V, \chi_U) = \dim_\C \Hom_{\C G}(U, V)$ \\ +Beweis: $\dim_\C \Hom_{\C G}(U, V) \mathop{=}\limits^\text{6.1.12} \dim_\C (\Hom_\C(U, V))^G \mathop{=}\limits^\text{6.1.11} = \frac{1}{\abs{G}} \sum\limits_{g \in G} \chi_{\Hom_\C(U,V)}(g)$ \\ +$ \mathop{=}\limits^\text{6.1.8} \frac{1}{\abs{G}} \sum\limits_{g \in G} (\overline{\chi_{U}(g)} \chi_V(g)) = (\chi_V, \chi_U)$ \qed + +Bemerkung: $\Hom_{\C G}(S_1^{\oplus \nu_1} \oplus \ldots \oplus S_r^{\oplus \nu_r}, S_1^{\oplus \mu_r} \oplus \ldots \oplus S_r^{\oplus \mu_r}) + = \bigoplus\limits_{i=1}^r \Hom_{\C G}(S_i^{\oplus \nu_i}, S_i^{\oplus \mu_i}) + \cong M_{\mu_i \times \nu_i}(\C) $ \\ + $ \dim = \sum \mu_i \nu_i$ + +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. + \end{document} diff --git a/standard.tex b/standard.tex index 4023dd4..5c7f067 100644 --- a/standard.tex +++ b/standard.tex @@ -95,6 +95,8 @@ \DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\Irr}{Irr} \DeclareMathOperator{\tr}{tr} % Spur +\DeclareMathOperator{\Cl}{Cl} + \DeclareMathOperator{\Reid}{Reid} \DeclareMathOperator{\Liid}{Liid}