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\fancyhead[L]{\bf\large PMATH 445/745: Representation Theory\\ Assignment 2}
\fancyhead[R]{\bf\large Winter 2025  \\Due Friday, Jan 24 at 7:00 PM}
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\begin{document}
	\noindent In the problems below, $G$ is a finite group. You may assume $F=\C$.
	
	\begin{enumerate}[{\bf Q1.}]
		\item (Copy of \textbf{A1Q5}.) Let $V$ and $W$ be finite-dimensional $F$-vector spaces.
		\begin{enumerate}
			\item Let $z\in V\otimes W$. Prove that there exist linearly independent $v_1,\ldots,v_n \in V$ such that $z = \sum_{i=1}^n v_i\otimes w_i$ for some $w_i\in W$.
			\item Let $V_0\subset V$ and $W_0\subset W$ be subspaces. Show that there is a unique injective linear map $T\colon V_0\otimes W_0 \to V\otimes W$ such that $T(v_0\otimes w_0) = v_0\otimes w_0$. %[Note: The two occurrences of the symbol $v_0\otimes w_0$ have two meanings. The first is an element of the tensor product space $V_0 \otimes W_0$; the second is an element of $V\otimes W$. A byproduct of this exercise is that there is no ambiguity here: The map $T$ allows us to view $V_0\otimes W_0$ as a subspace of $V\otimes W$ in a canonical way---so we can identify both instances of $v_0\otimes w_0$ with each other.]
			\item True or false? Every subspace of $V\otimes W$ is of the form $V_0\otimes W_0$ for some subspaces $V_0\subset V$ and $W_0\subset W$. [Note: Here $V_0\otimes W_0$ is being viewed as a subspace of $V\otimes W$ via the embedding given in part (b).]
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
			\item
		\end{enumerate}
		
		\pagebreak
		
		\item Let $X$ and $Y$ be finite $G$-sets. In this problem, view $X\times Y$ as a $G$-set with action given by $g(x,y) = (gx,gy)$.
		\begin{enumerate}
			\item Establish the following isomorphisms of $G$-modules:
			\begin{enumerate}[(i)]
				\item $F\langle X\times Y\rangle \cong F\langle X\rangle \otimes F\langle Y\rangle$.
				\item $F\langle X\rangle^* \cong F\langle X\rangle$.
				\item $\Hom(F\langle X\rangle, F\langle Y\rangle) \cong F\langle X \times Y\rangle$.
			\end{enumerate}
			\item Deduce that $\dim \Hom_G(F\langle X\rangle, F\langle Y\rangle)$ is equal to the number of $G$-orbits in $X\times Y$.
			\item Determine $\dim \Hom_{S_n}(V,V)$, where $V$ is the defining representation of $S_n$.
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\begin{enumerate}
		\item 
		\begin{enumerate}[(i)]
			\item 
			\item 
			\item 
		\end{enumerate}
		\item 
		\item
		\end{enumerate}
		
		\pagebreak
		
		\item Let $V$ be an $FG$-module and let $\chi \colon G\to F^\times$ be a one-dimensional representation of $G$. Put $V_\chi = \{ v\in V \colon gv = \chi(g)v \text{ for all } g\in G \}$ and note that this is a $G$-submodule of $V$. Let $F_\chi$ denote the $G$-module determined by $\chi$; that is, $F_\chi = F$ and the action of $g\in G$ on $a\in F$ is given by $g\cdot a = \chi(g)a$.
		\begin{enumerate}
			\item Show that $\Hom_G(F_\chi,V) \cong V_\chi$ as vector spaces.
			\item Determine $\dim \Hom_G(F_\chi, V_\reg)$ if $\chi$ is the trivial representation of $G$.
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
		\end{enumerate}
		
		\pagebreak
		
		\item Let $V$ and $W$ be finite-dimensional $G$-modules. Prove or disprove:
		\begin{enumerate}
			\item $V^*$ is irreducible if and only if $V$ is irreducible.
			\item $V\otimes W$ is irreducible if and only if $V$ and $W$ are irreducible.
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
		\end{enumerate}
		
		\pagebreak
	\end{enumerate}
\end{document}