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\title{Math 668 Problem Set 7: \\ Due Monday, October 31}





\begin{document}

\maketitle
\thispagestyle{empty}

%
%Work out S_3

\begin{Problem}
The point of this exercise is to work through the representation theory of the symmetric $S_3$ and see that all of the claims we have made in class are true.
I'll start by telling you the simple representations of $S_3$; there are three of them. One of them (called the trivial) is a one-dimensional representation with $\rho(g) = 1$ for all $g$. 
One of them (called the sign) is a one dimensional representation with $\rho(g) = \text{sign}(g) \in \{ 1, -1 \}$. And one of them is the two dimensional representation of $S_3$ on $\begin{sbm} 1\\1\\1 \end{sbm}^{\perp} \subset \RR^3$.

\begin{enumerate}
\item Describe the conjugacy classes of $S_3$. Check that there are also three of them.
\item Describe the characters of $S_3$. Check that they give an orthogonal basis for conjugacy invariant functions on $S_3$.
\item Write out matrices $\rho(g)$ for the actions on each of these representations, so that the matrices are orthogonal. Check that the functions $g \mapsto \sqrt{\dim V} \rho_V(g)_{ij}$ give an orthonormal basis for functions on $S_3$.
\end{enumerate}
\end{Problem}

\begin{Problem}
The group $O(2)$ is the group of $2 \times 2$ real orthogonal matrices. Every matrix in $O(2)$ is either of the form $\begin{sbm} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \\ \end{sbm}$ or $\begin{sbm} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \\ \end{sbm}$.  In this problem, we will work out the continuous representation theory of $O(2)$ in $\GL_n(\CC)$.

The subgroup of matrices in $O(2)$ with determinant $1$ is called $SO(2)$. (In other words, $SO(2)$ is the group of rotations.)
We showed in class that every continuous representation of $SO(2)$ is a direct sum of representations of the form $\rho_k : \begin{sbm} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \\ \end{sbm} \mapsto [ (\cos \theta + i \sin \theta)^k]$ for $k \in \ZZ$.

Let $V$ be a continuous complex representation of $O(2)$. Write $V = \bigoplus_k V_k$ where $SO(2)$ acts $V_k$ by  $\rho_k$.

\begin{enumerate}
\item Let $\sigma$ be a matrix in $O(2)$ with determinant $-1$ (in other words, a reflection). Show that $\sigma$ maps $V_k$ to $V_{-k}$.
\item Show that $V_0$ is a $O(2)$-subrepresentation of $V$ and, for $k>0$, show that $V_k \oplus V_{-k}$ is an $O(2)$-subrepresentation of $V$.
\item Show that $V_0$ is a direct sum of copies the following two representations: The $1$-dimensional trivial representation given by $\rho(g)=1$ for all $g\in O(2)$, and the $1$-dimensional representation where $\rho(g)=\det(g)$.
\item Show that $V_k \oplus V_{-k}$ is a direct sum of copies of the $2$-dimensional representation given by 
\[\begin{array}{rcl}\rho \left( \begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}  \right) &=& \begin{bmatrix}  (\cos \theta + i \sin \theta)^k & 0 \\ 0 &  (\cos \theta + i \sin \theta)^{-k} \end{bmatrix}  \\[0.4 cm]
  \rho \left( \begin{bmatrix} \sin \theta & \cos \theta \\ \cos \theta & - \sin \theta \\ \end{bmatrix}  \right) &=& \begin{bmatrix}  0 &  (\cos \theta + i \sin \theta)^{-k}\\ (\cos \theta + i \sin \theta)^k & 0 \\  \end{bmatrix} \\
\end{array} . \]
\end{enumerate}
Since you have shown that an arbitrary representation is the direct sum of the $1$- and $2$-dimensional representations listed here, you have found all representations of $O(2)$.
\end{Problem}

\begin{Problem}
Let $\kappa$ be an infinite field of characteristic $0$. Let $G$ be the group of matrices of the form $\begin{sbm} t & u \\ 0&1 \end{sbm}$, with $t \in \kappa^{\times}$ and $u \in \kappa$. (As an abstract group, $G \cong \kappa^{\times} \ltimes \kappa^+$.)
Let $\rho : G \to \GL(V)$ be an algebraic representation, meaning representations whose matrix entries are in $k[t, t^{-1}, u]$.

You have already shown, and may assume:
\begin{itemize}
\item That $V$ decomposes as $V = \bigoplus V_k$, so that $\begin{sbm} t&0 \\ 0&1 \end{sbm}$ acts on $V_k$ by $t^k$
\item That there is a nilpotent matrix $N$ such that $\rho \left( \begin{sbm} 1&u \\ 0&1 \end{sbm} \right) = \sum_{k=0}^{\infty} \tfrac{u^k N^k}{k!}$
\end{itemize}

\begin{enumerate}
\item Show that $N$ maps $V_k$ to $V_{k+1}$.
\item Given a finite dimensional vector space $V$ with a direct sum decomposition $V = \bigoplus V_k$, and a nilpotent operator $N$ mapping $V_k$ to $V_{k+1}$, show that there is a corresponding algebraic representation of $G$.
\end{enumerate}
\end{Problem}



\end{document}