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\fancyhead[L]{\bf\large PMATH 445/745: Representation Theory\\ Assignment 4}
\fancyhead[R]{\bf\large Winter 2025  \\Due Friday, Feb 7 at 7:00 PM}
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\begin{enumerate}[{\bf Q1.}]
	\item The \textbf{character} of a $\C C_n$-module $(V,\rho)$ is the function $\chi_\rho \colon G\to\C$  defined by $\chi_\rho(g)=\tr(\rho(g))$. Prove that two $\C C_n$-modules $(V,\rho)$ and $(W,\sigma)$ are isomorphic if and only if $\chi_\rho = \chi_\sigma$ as functions. [Hint: Use the isotypic decomposition of $V$ to diagonalize $\rho(a)$. Warm up with $n=2$.]
	
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	\textit{Solution:}
	
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	\item Assume $n\geq3$. In this problem you will determine the isotypic decomposition of the defining representation $V=\C^n$ of $S_n$. Let
	\[ U=\spn\{(1,1,\ldots,1)\} \qandq W=\{(x_1,\ldots,x_n)\in\C^n \colon x_1+\cdots+x_n=0\}. \]
	\begin{enumerate}
		\item Show that $U$ and $W$ are $S_n$-invariant and that $V=U\oplus W$.
		\item Let $e_1,\ldots,e_n$ denote the standard basis vectors of $\C^n$. Let $w$ be a nonzero vector in $W$. Show that $e_1 - e_i \in \spn\{\pi w \colon \pi \in S_n\}$ for all $i=2,\ldots,n$. Deduce that $W$ has no proper nonzero $S_n$-invariant subspaces.
	\end{enumerate}
	
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	\textit{Solution:}
	
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	\item In this problem you will determine $\Irr_\C(S_4)$.
	\begin{enumerate}
		\item Prove that the trivial representation and the alternating representation are the only one-dimensional representations of $S_4$.
		\item Prove that the tensor product of an irreducible representation and a one-dimensional representation is irreducible. (Compare A2 Q4.)
		\item Determine the number and dimensions of the irreducible representations of $S_4$.
		\item Figure out (or look up) how to realize $S_3$ as a quotient of $S_4$. Use this to produce an irreducible two-dimensional representation of $S_4$.
		\item Describe $\Irr_\C(S_4)$.
	\end{enumerate}
	
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	\textit{Solution:}
	
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