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

\noindent
{\large \textmd{Homework 5}}

\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 13th, 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 Reducibility, unitary representations.} }

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

\begin{enumerate}
\item Prove Lemmas 3.1.24, 3.1.25 in the textbook. See the proof of Lemma 3.1.23 for inspiration.
\item Let $G$ be a finite group, $\mathcal{R}$ the collection of all finite dimensional complex representations of $G$. Show that $V\sim V'$ if and only if $V$ is equivalent to $V'$ defines an equivalence relation on $\mathcal{R}$.
\item Exercise 3.6 in the textbook, including the following additional problem: {\sl Ex. $3.6.3$: show that $\varphi^{\chi}$ is irreducible if and only if $\varphi$ is irreducible}.
\item Exercise 3.8 in the textbook. 
\item Let $\rho:G\rightarrow \GL(V)$ be a representation of the finite group $G$.
\begin{enumerate}
\item Let $U,W\subset V$ be subrepresentations. Show that $U\cap W$ and $U+W$ are subrepresentations.
\item Let $U,W\subset V$ be irreducible subrepresentations. Prove that either $U\cap W=\{0_{V}\}$ or $U=W$.
\end{enumerate}
\item Let $X\in M_{n}(\C)$ be the matrix
	\[ X = [e_{2}\; e_{3}\; \cdots e_{n}\; e_{1}]\]
Here $e_{i}\in \C^{n}$ is the $i^{th}$ standard basis vector. Consider the map
	\[ \rho:\Z/n\Z \rightarrow \GL_{n}(\C)\;,\; \overline{j}\mapsto X^{j}\]
\begin{enumerate}
\item  Verify that $\rho$ defines a homomorphism. ({\sl Hint: utilise the standard representation $\varphi:S_{n}\rightarrow \GL_{n}(\C)$.})
\item Show that $\rho$ decomposes as a direct sum of $n$ inequivalent degree $1$ representations. ({\sl Hint: Exercise 3.4 will be useful here; also, it may help to experiment with the $n=5$ case})
\end{enumerate}
\end{enumerate}












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