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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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\fancyhead[L]{\bf\large PMATH 445/745: Representation Theory\\ Assignment 7}
\fancyhead[R]{\bf\large Winter 2025  \\Due Friday, Apr 4 at 7:00 PM}
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\begin{document}	
	\begin{enumerate}[{\bf Q1.}]
		\item Let $V\in\Irr_\C(G)$ and set $n=\dim_\C(V)$. The exercises below give a proof of Proposition 26.22.
		\begin{enumerate}
			\item Show that $\dim_\R \End_{\R G}(V_\R) = \dim_\C\End_{\C G}(V\oplus \overline{V})$.
			\item Assume $V$ cannot be defined over $\R$. Show that $V_\R$ must be irreducible. [Hint: Go for a contradiction and consider $(V_\R)_\C$.]
			\item Assume $\chi_V$ is not real-valued.
			\begin{enumerate}[(i)]
				\item Show that $\dim_\R \End_{\R G}(V_\R)=2$ hence deduce that $\End_{\R G}(V_\R) \cong \C$. [Hint: Lemma 23.8. For the last bit, note that $\End_{\R G}(V_\R)$ contains a copy of $\C$ since $V$ is a $\C$-vector space.]
				\item Explain why $V_\R$ is irreducible and show that the corresponding component in the Wedderburn decomposition of $\R G$ is $M_n(\C)$.
			\end{enumerate}
			\item Assume $\chi_V$ is real-valued.
			\begin{enumerate}[(i)]
				\item Show that $\dim_\R \End_{\R G}(V_\R) = 4$. [Hint:  Lemma 23.6.]
				\item Assume that $V$ can be defined over $\R$; say $V=W_\C$ with $W\in\Irr_\R(G)$. Show that $\End_{\R G}(W)=\R$ and that the corresponding Wedderburn component in $\R G$ is $M_n(\R)$. [Hint: Consider $\C \otimes \End_{\R G}(W)$.]
				\item Assume that $V$ cannot be defined over $\R$. By part (b), $V_\R$ is irreducible. Show that $\End_{\R G}(V_\R)=\bH$ and that the corresponding Wedderburn component is $M_{\frac n2}(\bH)$. [Hint: Frobenius.]
			\end{enumerate}
		\end{enumerate}
		
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item
		\begin{enumerate}
			\item Determine the Wedderburn decompositions of $\C A_4$, $\C A_5$, $\R A_4$ and $\R A_5$. [You are free to look up the character table of $A_4$. It appears in Q2a of Sample Test 2.]
			\item Give a list of all semisimple $\R$-algebras of dimension 4 (up to isomorphism). For each algebra $A$ in your list, either find a finite group $G$ such that $\R G \cong A$ or else prove that no such group exists.
			\item Same problem as (b) but over $\C$ and of dimension 10.
		\end{enumerate}
		[Note: For (b) and (c) give an individual reason for why $A$ cannot be a group ring. Do not argue along the lines of ``The only groups of order $N$ are (...), and their groups rings are (...)."]
		
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item Let $H \leq G$ and suppose that $\chi_1,\ldots,\chi_k$ are the irreducible characters of $G$. Let $\psi$ be an irreducible character of $H$. Show that if $\ind_H^G \psi = \sum_{i=1}^k d_i \chi_i$ then $\sum_{i=1}^k d_i^2 \leq (G:H)$.
		
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item Let $G=S_4$ and $H=\langle(1~2), (3~4)\rangle\cong S_2\times S_2$. Let $\psi$ be the trivial character of $H$.
		\begin{enumerate}
			\item Calculate $\langle\ind_H^G \psi, \chi\rangle$ for $\chi \in \{\chi_\triv,\chi_\sgn,\chi_\std\}$.
			\item Determine the isotypic decomposition of $\ind_H^G(\triv)$.
		\end{enumerate}
		
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\pagebreak
		
		\item (Bonus) Give a ``direct'' (i.e. no FS indicators, etc.) argument that shows the two-dimensional complex irrep of $Q_8$ cannot be defined over $\R$.
		
	\end{enumerate}
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