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

\noindent
{\large \textmd{Homework 4}}

\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 6th, 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 Subrepresentations, irreducible representations} }

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

\begin{enumerate}
\item Exercise 3.3, 3.4 in the textbook. ({\sl You've already done 3.3 (1) in Homework 3 so you can skip it.})
\item Let $V$ be a vector space over $\C$, and let $U, W\subset V$ be subspaces satisfying the following properties:
\begin{itemize}
\item $V=U+W = \{u+w\;|\;u\in U, w\in W\}$,
\item $U\cap W=\{0_{V}\}$.	
\end{itemize}
Show that, for every $v\in V$, there exists unique $u\in U$, $w\in W$ such that $v=u+w$. We say that $V$ is the {\bf (internal) direct sum of $U, W$}.
\item Let $V_{1}, V_{2}$ be vector spaces over $\C$. The {\bf (external) direct sum of $V_{1},V_{2}$} is the vector space $V_{1}\oplus V_{2}$ defined as follows:
\begin{itemize}
\item As a set $V_{1}\oplus V_{2} = V_{1}\times V_{2}$, the Cartesian product.
\item Define addition component-wise: $(u_{1}, u_{2}) + (v_{1},v_{2}) = (u_{1}+v_{1},u_{2}+v_{2})$, for any $(u_{1},u_{2}), (v_{1},v_{2})\in V_{1}\times V_{2}$.
\item Define scalar multiplication component-wise: $c(u_{1},u_{2}) = (cu_{1},cu_{2})$, for any $c\in \C$, $(u_{1},u_{2})\in V_{1}\times V_{2}$.
\item The zero vector is $(0_{V_{1}}, 0_{V_{2}})$.
\end{itemize}
Let $V$ be a vector space over $\C$, $U,W$ subspaces of $V$ satisfying the properties in the previous Exercise. Give a linear map
	\[T:U\oplus W\rightarrow V\;,\; (u,w)\mapsto T(u,w) \]
	that is an isomorphism of vector spaces. ({\sl Hint: given two vectors in $V$ how can you produce a third?})
\item Let $G=S_{3}$, the symmetric group on $3$ letters, and $\C^{3}$ be the standard representation.
\begin{enumerate}
\item Show that the subrepresentation $W=\nul([1\;1\;1])=\left\{\begin{bmatrix}a\\b\\c\end{bmatrix}\;\vert\; a+b+c=0\right\}$ is irreducible.
\item Show that $\C^{3}\simeq \C \oplus W$. Here $\C$ is the degree 1 trivial representation.
\end{enumerate}


\item Let $\rho : G\rightarrow \GL_{2}(\C)$ be a degree 2 representation of $G$. Suppose that $\rho\simeq \rho_{1}\oplus \rho_{2}$, where $\rho_{1},\rho_{2}$ are degree $1$ representations of $G$. Show that $\rho_{g}\rho_{h} = \rho_{h}\rho_{g}$, for all $g,h\in G$.
\end{enumerate}












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