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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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%% 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
		\begin{enumerate}
			\item Determine the character table of $A_5$.
			\item Deduce that $A_5$ is a simple group.
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item Let $\chi$ be a character of $G$. The \textbf{Frobenius--Schur indicator} of $\chi$ is the number
		\[ \ep(\chi) = \frac{1}{|G|} \sum_{g\in G} \chi(g^2). \]
		Assume $V$ is an irreducible $\C G$-module and let $\chi=\chi_V$.
		\begin{enumerate}
			\item Prove $\mult(V_\triv, V\otimes V)$ is either 1 or 0 depending on whether $V\cong V^*$ or not.
			\item Express $\ep(\chi)$ in terms of the characters of $\Sym^2(V)$ and $V\otimes V$.
			\item Hence show that
			\[ \ep(\chi) = \begin{cases} \pm1 & \text{if and only if $\chi_V$ is real valued} \\ 0 & \text{otherwise.} \end{cases} \]
			\item Explain briefly how the space of bilinear forms on $V$ can be identified with $(V\otimes V)^*$. Deduce that the space of $G$-invariant bilinear forms on $V$ (as defined in A3Q3) can be identified with $(V^* \otimes V^*)^G \cong \Sym^2(V^*)^G \oplus \Alt^2(V^*)^G$.
			\item Show that if there exists a non-zero $G$-invariant bilinear form $B$ on $V$ then it must belong to either $\Sym^2(V^*)^G$ (in which case it is a symmetric form, i.e. $B(x,y)=B(y,x)$) or else it must belong to $\Alt^2(V^*)^G$ (in which case it is a skew-symmetric form, i.e. $B(x,y)=-B(y,x)$).
			
			[Hint: What is $\dim\,(V^*\otimes V^*)^G$?]
			\item Show that $c_V$ from A3Q3 is the Frobenius--Schur indicator of $\chi_V$. Hence conclude that
			\[ \ep(\chi_V) = \begin{cases} 1 & \text{if there exists a nonzero symmetric $G$-invariant bilinear form on $V$;} \\ -1 &\text{if there exists a nonzero skew-symmetric $G$-invariant bilinear form on $V$;} \\ 0 & \text{if $V$ has no nonzero $G$-invariant bilinear forms.}\end{cases} \]
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item 
		\begin{enumerate}
			\item For each partition $\lambda\vdash 4$, match up the Specth module $S_\lambda$ with the appropriate irreducible representation $V\in\Irr_\C(S_4)$ that you determined in A4Q3.
			\item Let $V$ be the standard representation of $S_4$. Determine the isotypic decomposition of $\Sym^2(V)$. Express your answer in terms of the Specht modules $S_\lambda$ with $\lambda\vdash 4$.
			\item Determine the dimension of the Specht module $S_{(n-1,1)}\in\Irr_\C(S_n)$ by using:
			\begin{enumerate}
				\item The Hook Length formula.
				\item The Frobenius character formula.
			\end{enumerate}
			\item Let $V\in\Irr_\C(S_n)$. Prove that $\chi_V$ is integer-valued by using:
			\begin{enumerate}
				\item The Specht module construction.
				\item The Frobenius character formula.
			\end{enumerate}
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		%% Note: There is a bonus Q4 in the posted PDF.
	\end{enumerate}
\end{document}