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\fancyhead[L]{\bf\large PMATH 445/745: Representation Theory\\ Assignment 5}
\fancyhead[R]{\bf\large Winter 2025  \\Due Friday, Mar 7 at 7:00 PM}
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\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\bH}{\mathbb{H}}
\newcommand{\Fun}{\mathcal{F}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\triv}{\mathrm{triv}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\std}{\mathrm{std}}
\newcommand{\reg}{\mathrm{reg}}
\newcommand{\Irr}{\mathrm{Irr}}
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%% header
\pagestyle{fancy}
\fancyhead[L]{\bf\large PMATH 445/745: Representation Theory\\ Assignment 5}
\fancyhead[R]{\bf\large Winter 2025  \\Due Friday, Mar 7 at 7:00 PM}
%\fancyfoot[C]{Page \thepage\ of \pageref*{LastPage}}
\setlength{\headheight}{35pt}

\begin{document}	
	\begin{enumerate}[{\bf Q1.}]
		\item For $a,b\in\R^\times$, define the generalized quaternion algebra $\bH_{a,b}$ to be the 4-dimensional $\R$-vector space with basis $1,i,j,k$ and multiplication satisfying
		\[ i^2=a, \quad j^2=b \qandq ij=-ji=k. \]
		Note that the Hamilton quaternion algebra is $\bH=\bH_{-1,-1}$.
		\begin{enumerate}
			\item \textit{Do not submit.} Convince yourself that the above really defines an $\R$-algebra. In particular, what are the products $ik$, $ki$, $jk$, $kj$? Show that $k^2=-ab$.
			\item Show that there are isomorphisms of $\R$-algebras $\bH_{a,b} \cong \bH_{b,a}$ and $\bH_{u^2a,v^2b} \cong \bH_{a,b}$ for all $u,v\in\R^\times$. Hence deduce that $\bH_{a,b}$ is isomorphic to one of $\bH_{1,1}$, $\bH_{1,-1}$ and $\bH_{-1,-1}$.
			\item Show that $\bH_{1,1} \cong \bH_{1,-1} \cong M_2(\R)$ and that $\bH_{-1,-1} \not\cong M_2(\R)$.
		\end{enumerate}
		
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item Let $R=F[x]/(f(x))$, where $F$ is a field and $\deg f\geq1$. Suppose $f(x) = p_1(x)^{a_1} \cdots p_k(x)^{a_k}$ is the factorization of $f$ into distinct irreducibles $p_i(x)\in F[x]$. Set $S_i := F[x]/(p_i(x))$.
		\begin{enumerate}
			\item Show that $S_i$ is a simple $R$-module.
			\item Show, conversely, that every simple $R$-module is isomorphic to some $S_i$.
			\item Conclude that there are $k$ distinct simple $R$-modules up to isomorphism, and representatives for the isomorphism classes are given by $S_i$ for $1\leq i\leq k$.
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item 
		\begin{enumerate}
			\item Prove that $\R C_n \cong \R[x]/(x^n-1)$ as rings.
			\item Hence deduce:
			\begin{enumerate}
				\item If $n$ is odd, $\Irr_\R(C_n)$ consists of the trivial representation and $\frac{n-1}{2}$ two-dimensional representations.
				\item If $n$ is even, $\Irr_\R(C_n)$ consists of two one-dimensional representations and $\frac{n-2}{2}$ two-dimensional representations.
			\end{enumerate}
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item Let $M$ be an $R$-module and let $A,B,$ and $C$ be submodules of $M$ such that $C\subset A$. Prove:
		\begin{enumerate}
			\item $A\cap(B+C) = (A\cap B)+C$.
			\item If there is a submodule $C'$ such that $M=C\oplus C'$ then there is a submodule $C''$ such that $A=C\oplus C''$.
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
	\end{enumerate}
\end{document}