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\title{Math 668 Problem Set 6: \\ Due Friday, October 14}





\begin{document}

\maketitle
\thispagestyle{empty}

\begin{Problem}
Let $S_3$ act on $\RR^3$ by permuting the coordinate axes. The subspace $\RR \begin{sbm} 1\\1\\1 \end{sbm}$ is clearly a subrepresentation, so by our results in class, there is another representation $V$ with $\RR^3 = \RR \begin{sbm} 1\\1\\1 \end{sbm} \oplus V$. 
\begin{enumerate}
\item Describe $V$ explicitly.
\item Give explicit matrices for the action of $S_3$ on $V$.
\end{enumerate}
\end{Problem}

\begin{Problem}
Let $G$ be a finite group and let $X$ be a finite set on which $G$ acts. Let $V$ be the vector space of $\CC$-valued functions on $X$, with the obvious action of $G$.
\begin{enumerate}
\item Show that $\dim V^G$ is the number of orbits of $G$ on $X$.
\item Describe $\chi_V(g)$ in terms of the action of $g$ on $X$. For those who have had a bit of group theory, what does the identity $\dim V^G = \tfrac{1}{|G|} \sum_{g \in G} \chi_V(g)$ translate to in this case?
\item Show that $\dim \Hom_G(V,V)$ is the number of orbits of $G$ on $X \times X$.
\end{enumerate}
Let $V = \bigoplus U_i^{\oplus c_i}$ be the decomposition of $V$ into irreducible components; we call $\sum c_i$ the \newword{number of irreducible components}.
\begin{enumerate}[resume]
\item  Show that the only case where there is just one irreducible component is when $|X| = 1$, and give a criterion for when there are two irreducible components. 
\end{enumerate}
\end{Problem}

\begin{Problem} 
Here is a linear algebra lemma that we will need soon: Let $V$ be a vector space and let $P_1$, $P_2$, \dots, $P_k$ be linear maps $V \to V$ satisfying
\[ \begin{array}{rcl}
P_i^2 &=& P_i \\
P_i P_j &=& 0 \ \text{for}\ i \neq j \\
\sum_{i=1}^k P_i &=& \text{Id} \\
\end{array}. \]
Show that $V$ can be decomposed as $\bigoplus_{i=1}^k V_i$ such that $P_i$ is the projection onto $V_i$.
\end{Problem}

\begin{Problem}
Let $k$ be a field of characteristic zero and let $k^+$ be its additive group. We say that a representation $\rho : k^+ \to \GL_n(k)$ of $k^+$ is \newword{algebraic} if the entries of the matrix $\rho(t)$ are polynomials in $t$. In this problem, we will describe all algebraic representations of $k^+$. The analogous computation for $k^{\times}$ will be done in class.
\begin{enumerate}
\item Show that there is a nilpotent matrix $N$ such that $\rho(t) = \sum_{k=0}^{\infty} \tfrac{t^k N^k}{k!}$.
\item Conversely, if $N$ is any nilpotent matrix, show that the above formula gives a polynomial representation of $k^+$.
\end{enumerate}
\end{Problem}

\end{document}