\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm,enumerate,fancyhdr,hyperref,parskip,lastpage}
\usepackage[left=2.6cm, right=2.6cm, top=1.5cm, includehead, includefoot]{geometry}

%% header
\pagestyle{fancy}
\fancyhead[L]{\bf\large PMATH 445/745: Representation Theory\\ Assignment 1}
\fancyhead[R]{\bf\large Winter 2025  \\Due Friday, Jan 17 at 7:00 PM}
%\fancyfoot[C]{Page \thepage\ of \pageref*{LastPage}}
\setlength{\headheight}{35pt}

%% macros
\newcommand{\C}{\mathbb{C}}
\newcommand{\Fun}{\mathcal{F}}


\begin{document}
	\noindent You may assume that all representations are over $F=\C$.
	
	\begin{enumerate}[{\bf Q1.}]
		
		\item Tell me a bit about yourself.
		\begin{enumerate}
			\item Have you taken any other 400-level (or equivalent) math courses? If so, which ones?
			\item What do you hope to get out of PMATH 445/745?
		\end{enumerate}
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
		\end{enumerate}
		
		\pagebreak
		
		\item Let $G$ be a finite group.
		\begin{enumerate}
			\item Let $X$ be a finite $G$-set and let $V$ be the induced permutation representation. Show that $\dim V^G$ is equal to the number of $G$-orbits in $X$. [A \textbf{$G$-set} is a set with a $G$-action. A \textbf{$G$-orbit} is a subset of $X$ of the form $\mathcal O_x = \{ gx \colon g\in G\}$ for some $x\in X$.]
			\item Let $V=\Fun(G,F)$ be the regular representation of $G$. Show that $V^G$ is the subspace of constant functions, and explain how $\dim V^G=1$ is consistent with part (a).
		\end{enumerate}
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
		\end{enumerate}
		
		\pagebreak
		
		\item Let $G=S_3$ and let $\sigma=(1~2)$ and $\tau=(1~2~3)$. Note that $\sigma$ and $\tau$ generate $G$. Let $H=\langle \sigma \rangle$ be the subgroup of $G$ generated by $\sigma$, and let $\rho \colon G \to GL(V)$ be the permutation representation induced by the action of $G$ on $G/H$.
		\begin{enumerate}
			\item Write down a basis for $V$ consisting of coset representatives for $G/H$. Then write down the corresponding matrices for $\rho(\sigma)$ and $\rho(\tau)$.
			\item Prove that $V$ is isomorphic to the defining representation of $S_3$.
		\end{enumerate}
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
		\end{enumerate}
		
		\pagebreak
		
		\item The group $C_2=\langle a \rangle$ has exactly two distinct one-dimensional representations over $\C$---namely, $\chi_\pm \colon C_2\to\C^\times$ given by $\chi_\pm(a^i) = (\pm1)^i$.
		\begin{enumerate}
			\item Show that every $\C C_2$-module $V$ can be decomposed as a direct sum $V = V_+ \oplus V_-$ of $G$-invariant subspaces where, for $v\in V_\pm$, $\rho(a^i)v = \chi_\pm(a^i)v$. [Hint: Eigenvectors?]
			\item The group $C_2$ acts on $V=M_n(\C)$ by $a\cdot A = A^T$. (Note that this is a linear action.) The decomposition $V=V_+\oplus V_-$ in this case is a familiar one. What is it? That is, describe as succinctly as possible the subspaces $V_\pm$ and the decomposition $A=A_+ + A_-$ of a matrix $A$ into the sum of matrices $A_\pm \in V_\pm$.
		\end{enumerate}
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
		\end{enumerate}
		
		\pagebreak
		
		\item 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 canonical embedding given in part (a).]
		\end{enumerate}
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
			\item 
		\end{enumerate}
		
		
	\end{enumerate}
\end{document}