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{\Large \textmd{MA434: Spring 2019}}

\noindent
{\large \textmd{Homework 6}}

\noindent
{\bf Contact:} \texttt{gwmelvin\makeatletter@\makeatother colby.edu}


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\noindent
\textbf{Some thoughts and advice:} 
\begin{itemize}
\item Please submit solutions to the following problems by {\bf Wednesday, March 20th, 1.10pm}. You can either submit your solution in class or leave it outside my office.
\item You should expect to spend several hours on homework sets. A lot of practice problem-solving is essential to understand the material and skills covered in class. Be organised and do not leave problem sets until the last-minute. Instead, get a good start on the problems as soon as possible. 
\item When approaching a problem think about the following: \textsl{do you understand the words used to state the problem? what is the problem asking you to do? can you restate the problem in your own words? have you seen a similar problem worked out in class? is there a similar problem worked out in the textbook? what results/skills did you see in class that might be related to the problem?}

If you are stuck for inspiration come to office hours, or send me an email. However, don't just ask for the solution - provide your thought process, the difficulties you are having, and ask a coherent question in complete English sentences. 
\item Form study groups - get together and work through problem sets. \textbf{This will make your life easier!} You must write your solutions \textsl{on your own} and \textsl{in your own words}.
\item You {\bf are not allowed} to use any additional resources (e.g. \texttt{stackoverflow.com}).  If you are concerned then please ask.
\end{itemize}
\hrule

\medskip
{\large {\bf $G$-morphisms} }

\medskip
{\sc Unless otherwise specified, all vector spaces will have scalar field $\C$.}

\begin{enumerate}
\item Let $T:V\rightarrow W$ be a $G$-morphism from $(\rho,V)$ to $(\phi,W)$. 
\begin{enumerate}
\item Prove that there exists a subrepresentation $U\subset V$ such that $V \simeq U\oplus \ker T$ and $U\simeq \im T$.
\item  Deduce that if $T$ is surjective then there exists a subrepresentation $U\subset V$ such that $U\simeq W$.
\end{enumerate}

\item Let $(\rho,V)$, $(\varphi, W)$ and $(\psi, U)$ be representations of $G$.
\begin{enumerate}
\item Show that 
	\[\Hom_{G}(\rho, \varphi) = \{ T:V \rightarrow W\;|\; \text{ $T$ is a $G$-morphism}\}\]
	 is a vector subspace of $\Hom(V,W) =\{ T:V\rightarrow W\;|\; \text{$T$ linear}\}$.
\item Let $T\in \Hom_{G}(\rho,\varphi)$, $S\in \Hom_{G}(\varphi,\psi)$. Show that $S\circ T \in \Hom_{G}(\rho,\psi)$.
\item Deduce that $\End_{G}(\rho) = \Hom_{G}(\rho,\rho)$ is a subring of $\End(V)$.
\end{enumerate}
{\bf In the remainder of the Exercises we will describe the subring $\End_{G}(\rho)$ explicitly, for some representations $\rho$.}
\item Consider the representation
	\[ \rho: \Z/3\Z \rightarrow \GL_{3}(\C)\;,\; \overline{j} \mapsto \begin{bmatrix} 0&0&1\\1&0&0\\0&1&0\end{bmatrix}^{j}\]
	In Homework 5, Exercise 6, you showed that $\C^3 = U_{1}\oplus U_{2}\oplus U_{3}$, with each $U_{i}=\Span(v_{i})\subset \C^{3}$ a degree $1$ subrepresentation, and $U_{1}, U_{2}, U_{3}$ are mutually inequivalent. In particular, $B=(v_{1},v_{2},v_{3})\subset V$ is a basis of common eigenvectors for all $\rho_{g}$, $g\in G$.
	\begin{enumerate}
	\item Let $T\in \End_{G}(\rho)$ be a $G$-morphism. Suppose that
		\[T(v_{1}) = a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3},\qquad a_{1},a_{2},a_{3}\in \C\].
		Show that $a_{2}=a_{3}=0$.
	\item Deduce that $[\rho_{g}]_{B}$ is diagonal, for every $g\in G$.
	\item Using the `{\sl matrix with respect to $B$}' map $\End(V)\rightarrow M_{3}(\C)\;,\; T\mapsto [T]_{B}$, show that there is an isomorphism of rings
	 	\[\End_{G}(\rho) \simeq \C\times \C\times \C.\]
	 Here $\C\times \C\times \C$ is the product ring with component-wise addition and multiplication.
	\end{enumerate}
\item Let $\rho:\Z/n\Z\rightarrow \GL_{n}(\C)$ be the representation from Homework 5, Exercise 6. Formulate and prove a conjecture identifying $\End_{G}(\rho)\simeq R$, where $R$ is a more familiar (commutative) ring. 
\end{enumerate}












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