\documentclass[12pt]{report}

\textheight 22cm
\textwidth 15.5cm
\oddsidemargin 0pt\evensidemargin 0pt
%\oddsidemargin 14pt\evensidemargin 0pt
%\topmargin -40pt
\topmargin-30pt
%\bottommargin0pt
\def\baselinestretch{1.1}
%\addtolength{\parskip}{1ex}
\jot=.5ex
%\parskip = 0.02in


\setlength\arraycolsep{2pt}



\usepackage{amssymb}
\usepackage{amsmath,bm}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsfonts}         
\usepackage{fancybox}   

%\usepackage[numbers]{natbib}

\usepackage{enumitem}

\usepackage{slashed}




\usepackage[usenames,dvipsnames]{xcolor}%http://en.wikibooks.org/wiki/LaTeX/Colors

\definecolor{darkgreen}{rgb}{0,0.4,0}
\definecolor{darkred}{rgb}{0.4,0,0}
\definecolor{darkblue}{rgb}{0,0,0.4}
\definecolor{lightblue}{rgb}{.6,.6,0.9}
\newcommand{\cobl}{\color{darkblue}}

\newcommand{\cor}{\color{red}}
\newcommand{\cog}{\color{darkgreen}}
\newcommand{\cob}{\color{black}}

\definecolor{uglybrown}{rgb}{0.8,  0.7,  0.5}

\def\mathmap#1#2#3#4{
#1 & \to &  #2 \cr 
#3 & \mapsto & #4 
}



\def\gG{\mathsf{G}}
\def\ii{{\bf i}}
\def\Ione{\mathbb{I}}
\def\UU{{\bf U}}
\def\HH{{\bf H}}
\def\pp{{\bf p}}
\def\aa{{\bf a}}
\def\qq{{\bf q}}
\def\eps{\epsilon}
\def\half{{1\over 2}}
\def\Tr{{{\rm Tr~ }}}
\def\tr{{\rm tr}}
\def\Re{{\rm Re\hskip0.1em}}
\def\Im{{\rm Im\hskip0.1em}}
\def\ppi{\boldsymbol{\pi}}
\def\pphi{\boldsymbol{\phi}}
%\def\pphi{\phi}
\def\grad{\vec \nabla}
\def\vB{\vec B}
\def\vE{\vec E}
\def\vA{\vec A}
\def\vAA{ \vec{\bf A}}
\def\vEE{{{\vec {\bf E}}}}
\def\vBB{{\vec {\bf B}}}

\def\CA{{\cal A}}
\def\CB{{\cal B}}
\def\CC{{\cal C}}
\def\CD{{\cal D}}
\def\CE{{\cal E}}
\def\CF{{\cal F}}
\def\CG{{\cal G}}
\def\CH{{\cal H}}
\def\CI{{\cal I}}
\def\CJ{{\cal J}}
\def\CK{{\cal K}}
\def\CL{{\cal L}}
\def\CM{{\cal M}}
\def\CN{{\cal N}}
\def\CO{{\cal O}}%AEL
\def\CP{{\cal P}}
\def\CQ{{\cal Q}}
\def\CR{{\cal R}}
\def\CS{{\cal S}}
\def\CT{{\cal T}}
\def\CU{{\cal U}}
\def\CV{{\cal V}}
\def\CW{{\cal W}}
\def\CX{{\cal X}}%AEL
\def\CY{{\cal Y}}
\def\CZ{{\cal Z}}

\def\IZ{\mathbb{Z}}
\def\IR{\mathbb{R}}
\def\eg{{\it e.g.}~}
\def\ie{{\it i.e.}~}
\def\up{\uparrow}
\def\down{\downarrow}


\def\bra#1{\left\langle#1\right|}
\def\ket#1{\left|#1\right\rangle}
\def\bbra#1{{\langle\langle}#1|}
\def\kket#1{|#1\rangle\rangle}
\def\vev#1{\left\langle{#1}\right\rangle}

\def\ketbra#1#2{ | #1 \rangle\hskip-2pt\langle #2|}



\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\({\left(}
\def\){\right)}

%%                              TABLEAUX.TEX
%%      This  macro file is for producing a ``Young Tableau'' which is
%%      an array of little squares sometimes used in mathematical physics.
%%      For instance, the command $\tableau{6 3 2}$ will produce a tableau
%%      with 6 squares in the top row, 3 in the next, and 2 in the last.
%%                                  OOOOOO
%%      This tableau will look like OOO    but made of squares instead of O's.
%%                                  OO
%%      Any number of rows may be present, each having a nonzero number of
%%      squares.
%%
%%      A tableau is math mode material, so use $ or $$ to enclose it.
%%
%%      The size and line-thickness of the little boxes are controlled by the
%%      dimension parameters --
%%              \tableauside=1.0ex              %(size)
%%              \tableaurule=0.4pt              %(line-thickness)
%%      Change them if you want.
%%
%%                                                      -- Doug Eardley 9/19/8%%
%%
\newdimen\tableauside\tableauside=1.0ex
\newdimen\tableaurule\tableaurule=0.4pt
\newdimen\tableaustep
\def\phantomhrule#1{\hbox{\vbox to0pt{\hrule height\tableaurule width#1\vss}}}
\def\phantomvrule#1{\vbox{\hbox to0pt{\vrule width\tableaurule height#1\hss}}}
\def\sqr{\vbox{%
  \phantomhrule\tableaustep
  \hbox{\phantomvrule\tableaustep\kern\tableaustep\phantomvrule\tableaustep}%
  \hbox{\vbox{\phantomhrule\tableauside}\kern-\tableaurule}}}
\def\squares#1{\hbox{\count0=#1\noindent\loop\sqr
  \advance\count0 by-1 \ifnum\count0>0\repeat}}
\def\tableau#1{\vcenter{\offinterlineskip
  \tableaustep=\tableauside\advance\tableaustep by-\tableaurule
  \kern\normallineskip\hbox
    {\kern\normallineskip\vbox
      {\gettableau#1 0 }%
     \kern\normallineskip\kern\tableaurule}%
  \kern\normallineskip\kern\tableaurule}}
\def\gettableau#1 {\ifnum#1=0\let\next=\null\else
  \squares{#1}\let\next=\gettableau\fi\next}

\tableauside=1.0ex
\tableaurule=0.4pt

%
             \tableauside=0.9ex              %(size)
             \tableaurule=0.8pt              %(line-thickness)
%



\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}


\def\parfig#1#2{
\parbox{#1\textwidth}
{\includegraphics[width=#1\textwidth]{#2}}
}



%%from  Rudro Rana Biswas
\usepackage[pagebackref,  % this puts links to the page numbers where refs appear
%pdftex, 
bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
\hypersetup{colorlinks=true, 
linkcolor=BrickRed, 
citecolor=Violet, 
filecolor=OliveGreen, 
urlcolor=RoyalBlue, 
filebordercolor={.8 .8 1}, 
urlbordercolor={.8 .8 0}}%http://en.wikibooks.org/wiki/LaTeX/Hyperlinks


%\usepackage{mathtools} % for inclusion arrow \xhookrightarrow{}


\renewcommand{\theequation}{\arabic{equation}}
\newif{\ifeq}           % defines a new condition @eq tested by the conditional \ifeq
\eqtrue                 % if uncommented, declares @eq to be true
%\eqfalse              % if uncommented, declares @eq to be false
%                                %
%                                % to use this, wrap text with the conditional, eg:
%                                %
%                                % \ifeq
%                                % SHOW THIS IFF \eqtrue HAS BEEN DECLARED
%                                % \fi
%
\def\answer#1
{
\ifeq
\textcolor{darkblue}{#1}
\fi
}
\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
Prof.~John McGreevy

{\Large\bf  Physics 220 Symmetries Fall 2024}\\
{\Large\bf Assignment 4 \answer{--~~~ Solutions} }
\end{center}

Comment: this problem set contains many problems, but most of them are simple and many of them are bonus problems.    


\noindent
\hfill {\bf Due 3:30pm { Thursday, October 24, 2024}} 


\hrule




\begin{enumerate}

\item {\bf Brain-warmer.} If I tell you that $\vev{\chi_R, \chi_R} \equiv {1\over |G|} \sum_{g \in G} \chi_R^\star(g) \chi_R(g)  = 2$
for some representation $R$ of a finite group $G$, do you know whether $R$ is irreducible?  Explain.



\item{\bf Brain-warmer.} 
Given two representations $A, B$ of a group $G$ (with carrier spaces $U$ and $V$ respectively) and an intertwiner between them
$$ \Lambda: U \to V, ~~~\Lambda A(g) = B(g) \Lambda \forall g \in G$$
show that $\ker \Lambda \subset U $ and $\Im \Lambda \subset V$ are 
invariant subspaces.  (Recall that this was the crucial ingredient in Schur's Lemma.)



\def\jj{{\bf j}}
\def\kk{{\bf k}}


\item {\bf Character table for the {quaternions}.}
Figure out the character table for the {quaternion group $Q_8$ (on page 11 of the lecture notes)} by whatever means necessary (don't look it up).




\item {\bf Irreps and conjugacy classes.}
\begin{enumerate}
Consider the object 
$$ S_\alpha^a \equiv {1\over n_\alpha}  \sum_{g \in C_\alpha} D^a(g) ,$$
a linear operator on $V_a$, the carrier space for an irrep of $G$.  $C_\alpha$ is a conjugacy class of $G$.
\item Show that $S_\alpha^a$ commutes with all the $D^a(g)$.


\item Use Schur's lemma to conclude that $S_\alpha^a = \lambda_\alpha^a \Ione_{a}$ and find $\lambda_\alpha^a$ in terms of familiar objects.
\label{part-b}



\item{} [Bonus problem] Conclude that ${\bf C}_\alpha {\bf P}^a = \lambda_\alpha^a {\bf P}^a $
where ${\bf P}^a  = {d_a \over |G| } \sum_{g \in G}\( \chi^a(g)\)^\star {\bf g}$ 
is the projector in the group algebra associated with the irrep $a$ and ${\bf C}_\alpha \equiv {1\over n_\alpha} \sum_{g \in C_\alpha} {\bf g}$ 
are elements of the center of the group algebra.


\end{enumerate}



\item {\bf Statistics of cycle lengths of permutations.}
[This is a bonus problem]


\begin{enumerate}

\item What fraction of permutations in $S_n$ have just one big cycle?  


\item What fraction of permutations in $S_{2n}$ have no cycle of length greater than $n$?  
What is the large-$n$ limit of this number?

[Hint: Count the number of ways to assign elements to such a cycle.]



\item What fraction of permutations in $S_n$ have no fixed points?  
What is the large-$n$ limit of this number?  

[Hint: One way to do it is to use inclusion-exclusion.  
Another way is to find a recursion relation for $D(n)$, the number of elements of $S_n$ with no fixed point.  
A third, trickier, way is to use our formula for cycle-lengths; to use this method, I had to pass through the grand canonical ensemble, \ie sum over $n$.]


\end{enumerate}

\item {\bf Completely unrelated riddles.}


\begin{enumerate}


\item There are 100 students in the group theory class.
The professor is tired of grading and decides to let the students grade each others' exams.
He arranges 100 boxes and uniformly randomly puts one exam in each box.  
The students must each open one box and grade the exam they find.  
What's the probability that no student grades their own exam?  




\item{} [I recommend this puzzle very highly.] The next year, there are again 100 students in the group theory class.
This time, for the final exam, the evil professor decides to assign a group project.  
He arranges 100 numbered boxes and uniformly randomly puts the name of one student in each box.  
Each student is allowed to open half the boxes.  
If every student finds their own name then all the students pass, otherwise they all fail.   
The students are not allowed to communicate with each other after the box-opening begins, but they are allowed to develop a strategy together beforehand.
What is a strategy that allows them all to pass with probability $> .3$?  



\item A group of $n$ comic artists is playing a game of exquisite corpse.  This means that each person draws one frame of the comic, and then passes it on to the next.  
If the order of artists is determined by a uniformly random element of $S_n$, what is the probability that the result is a comic with $n$ frames, each by a different artist?  




\end{enumerate}



\end{enumerate}

\end{document}
