Char table for d3

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Chapters/Chapter1.tex

@@ -14,9 +14,9 @@ \section{Introduction}
 Groups arise naturally as sets of symmetries of some object which are closed under composition and taking inverses.  For example, 
 \begin{enumerate}
 \item The \textbf{symmetric group} of degree $n$, $S_n$, is the group of all symmetries of the set $\{ 1, \ldots, n \}$.
-\item The \textbf{dihedral group} of order $2n$, $D_{2n}$, is the group of all symmetries of the regular $n$-gon in the plane.
+\item The \textbf{dihedral group} of order $2n$, $D_{n}$, is the group of all symmetries of the regular $n$-gon in the plane.
 \end{enumerate}
-In these two examples, $S_n$  acts on the set $\{ 1, \ldots, n \}$ and $D_{2n}$ acts on the regular $n$-gon in a natural manner. One may wonder more generally:  Given an abstract group $G$, which objects $X$ does $G$ act on?
+In these two examples, $S_n$  acts on the set $\{ 1, \ldots, n \}$ and $D_{n}$ acts on the regular $n$-gon in a natural manner. One may wonder more generally:  Given an abstract group $G$, which objects $X$ does $G$ act on?
 This is the basic question of representation theory, which attempts to classify all such $X$ up to isomorphism.
 
 %----------------------------------------------------------------------------------------
@@ -357,8 +357,8 @@ \section{Complete Reducibility}
 \end{enumerate}
 \end{note}
 
-\begin{example}[A $2$-dimensional irreducible representation]
-Let $G = D_6 = \{ \sigma, \tau | \sigma^3 = \tau^2 = e, \tau \sigma \tau^{-1} = \sigma^{-1} \}$. (Note that $D_6 \cong S_3)$.
+\begin{example}[A $2$-dimensional irreducible representation] \label{2d-irrep-d3}
+Let $G = D_3 = \langle \sigma, \tau | \sigma^3 = \tau^2 = e, \tau \sigma \tau^{-1} = \sigma^{-1} \rangle$. (Note that $D_3 \cong S_3)$.
 Consider the regular triangle centered at the origin with vertices
 \[(1,0), (-\frac{1}{2}, \frac{\sqrt{3}}{2}), (-\frac{1}{2}, - \frac{\sqrt{3}}{2}). \]
 We can let $\sigma$ act as rotation by $\frac{2 \pi}{3}$ and let $\tau$ act as reflection over the $x$-axis to obtain an action of $G$ on $\mathbb{C}^2$ given (under the standard basis) by the matrices
@@ -1183,9 +1183,9 @@ \section{Character Theory}
 \end{proof}
 
 \begin{example}
-Consider $G=D_6$.  We have seen three irreducible representations of $G$, namely the $1$-dimensional trivial representation, the $1$-dimensional alternating representation, and the $2$-dimensional irreducible representation $W$ constructed geometrically.  Then
-\[ |D_6| = 6 = 1^2 + 1^2 +2^2 \]
-so these are all of the irreducible representations of $D_6$ up to isomorphism.
+Consider $G=D_3$.  We have seen three irreducible representations of $G$, namely the $1$-dimensional trivial representation, the $1$-dimensional alternating representation, and the $2$-dimensional irreducible representation $W$ constructed geometrically in Example \ref{2d-irrep-d3}.  Then
+\[ |D_3| = 6 = 1^2 + 1^2 +2^2 \]
+so these are all of the irreducible representations of $D_3$ up to isomorphism.
 \end{example}
 
 \begin{defn}
@@ -1201,14 +1201,61 @@ \section{Character Theory}
 \end{align*}
 where the latter sum ranges over the conjugacy classes $[g]$ of $G$.
 
+\begin{defn}
+We define \textbf{the character table of } $G$ to be the table of complex numbers whose:
+\begin{itemize}\item rows are index by the isomorphism classes of irreducible representations of $G$, 
+\item columns are indexed by the conjugacy classes of $G$,
+\item $i,j$ entry is given by value of the character corresponding to row $i$ evaluated at the isomorphism class corresponding to column $j$.
+\end{itemize}
+\end{defn}
+\begin{note}
+It is also helpful to include the size of the conjugacy class associated with each column of the character table, so that we can easily compute the inner product of two characters by looking at the character table.
+\end{note}
+
+\begin{example}
+The conjugacy classes of $D_3 = \langle \sigma, \tau | \sigma^3 = \tau^2 = e, \tau \sigma \tau^{-1} = \sigma^{-1} \rangle$ are given by
+$\{e\}$, $\{ \sigma, \sigma ^2 \}$, and $\{\tau, \tau \sigma, \tau \sigma^2 \}$.  Therefore the character table of $D_3$ is given by 
+
+\begin{tabular}{ | l | c | c | c |}\hline 
+\multicolumn{4}{|c|}{Character table of $D_3$} \\ \hline
+Size of conjugacy class & 1 & 3 & 2 \\ \hline
+\multicolumn{4}{|c|}{} \\ \hline
+
+Conjugacy class representative $[g]$ & $[e]$ & $[\tau]$ & $[\sigma]$ \\ \hline
+$\chi_1$  ($1$-d trivial reprn) & $1$ & 1 & 1 \\ \hline
+$\chi_{\text{sgn}}$  ($1$-d sign reprn)  & 1 & -1 & 1 \\ \hline
+$\chi_W$ ($2$-d reprn obtained geometrically) & 2 & 0 & -1 \\
+\hline
+\end{tabular}
+\end{example}
+
+
 \begin{example}
-Let $G = D_8$.  Let $U_1, \ldots, U_r$ be the irreducible representations of $D_8$ with dimensions $d_1, \ldots, d_r$ respectively, where $U_1$ is the $1$-dimensional trivial representation.  Then 
+Let $G = D_4$.  Let $U_1, \ldots, U_r$ be the irreducible representations of $D_4$ with dimensions $d_1, \ldots, d_r$ respectively, where $U_1$ is the $1$-dimensional trivial representation.  Then 
 \[ d_2^2 + \ldots + d_r ^2 = |G| - d_1^2 = 8 - 1 = 7. \]
 There are two possibilities:  
 
 1. $r=8$, and $d_i = 1$ for all $ 1 \leq i \leq 8$. 
 
 2. $r=5$, and $d_2 = d_3 = d_4 = 1$, $d_5 = 2$.
 
-We saw in Example \ref{irr-char-of-d8} that $G$ has a two-dimensional irreducible representation, so in fact (2) holds.  The four 
+We saw in Example \ref{irr-char-of-d8} that $G$ has a two-dimensional irreducible representation, so in fact (2) holds.  The character table for $D_4$ is as follows:
+
+\begin{tabular}{ |l|l|l| }
+\hline
+\multicolumn{3}{ |c| }{Team sheet} \\
+\hline
+Goalkeeper & GK & Paul Robinson \\ \hline
+\multirow{4}{*}{Defenders} & LB & Lucas Radebe \\
+ & DC & Michael Duburry \\
+ & DC & Dominic Matteo \\
+ & RB & Didier Domi \\ \hline
+\multirow{3}{*}{Midfielders} & MC & David Batty \\
+ & MC & Eirik Bakke \\
+ & MC & Jody Morris \\ \hline
+Forward & FW & Jamie McMaster \\ \hline
+\multirow{2}{*}{Strikers} & ST & Alan Smith \\
+ & ST & Mark Viduka \\
+\hline
+\end{tabular}
 \end{example}

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+\newlabel{2d-irrep-d3}{{1.26}{9}{A $2$-dimensional irreducible representation}{thm.46}{}}
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