\documentclass[ignorenonframetext,11pt]{beamer}
\usepackage{amsmath,amssymb}
\usepackage{graphicx}
\usepackage{pgfarrows,pgfnodes}
\usepackage{url}
\usepackage{textcomp}
\usepackage[vcentermath]{youngtab}

\mode<presentation>
{
  \usetheme{Malmoe}
  \useinnertheme{circles}
  \setbeamertemplate{bibliography item}[text]
  
  \definecolor{artbasecolor}{rgb}{1,.5,0}
  \colorlet{artcolor}{artbasecolor!90!black}
  %\usecolortheme[named=artcolor]{structure}
  
  \setcounter{secnumdepth}{-1}
}

\title[FFTs for the Symmetric Group]{Beamer Class Demonstration}
\author[Eric Malm]{Eric~Malm}
\date{14 Feb 2005}
\institute{MATH 198: Math Forum\\Harvey Mudd College}

% replace 'placeholder' with own logo image
\pgfdeclareimage[height=1cm]{logo}{wcrest.jpg}
\logo{\pgfuseimage{logo}}

\Ylinethick 0.5pt

%% Macros for Bratteli diagrams
%% redefine numbering commands for specific uses
\newcommand{\irrIione}{1}
\newcommand{\irrIi}{\large\young(\irrIione)}

\newcommand{\irrIIione}{1}
\newcommand{\irrIIitwo}{2}
\newcommand{\irrIIi}{\large\young(\irrIIione\irrIIitwo)}

\newcommand{\irrIIiione}{1}
\newcommand{\irrIIiitwo}{2}
\newcommand{\irrIIii}{\large\young(\irrIIiione,\irrIIiitwo)}

\newcommand{\irrIIIione}{1}
\newcommand{\irrIIIitwo}{2}
\newcommand{\irrIIIithree}{3}
\newcommand{\irrIIIi}{\large\young(\irrIIIione\irrIIIitwo\irrIIIithree)}

\newcommand{\irrIIIiione}{1}
\newcommand{\irrIIIiitwo}{2}
\newcommand{\irrIIIiithree}{3}
\newcommand{\irrIIIii}{\large\young(\irrIIIiione\irrIIIiitwo,\irrIIIiithree)}

\newcommand{\irrIIIiiione}{1}
\newcommand{\irrIIIiiitwo}{2}
\newcommand{\irrIIIiiithree}{3}
\newcommand{\irrIIIiii}{\large\young(\irrIIIiiione,\irrIIIiiitwo,\irrIIIiiithree)}

\begin{document}

\begin{frame}
  \maketitle
\end{frame}

\begin{frame}
  \frametitle{Overview}
  \tableofcontents
\end{frame}

\section{Beamer Features}

\subsection{Structural Features}

\begin{frame}
  \frametitle{Structural Features}
  \begin{block}{Levels of Structure}
    \begin{itemize}
      \item usual \LaTeX\ \textbackslash{}section, \textbackslash{}subsection 
      commands
      
      \item `frame' environments provide slides
      
      \item `block' environments divide slides into logical sections
      
      \item `columns' environments divide slides vertically (example later)
      
      \item overlays (\`a la prosper) change content of slides dynamically
    \end{itemize}
  \end{block}
  
  \begin{example}[Overlay Alerts]
    On the first overlay, \alert<1>{this text} is highlighted (or \emph{alerted}).\\ On the second, \alert<2>{this text} is.
  \end{example}
\end{frame}

\subsection{Other Features}

\begin{frame}
  \frametitle{Other Features}

  \begin{block}{Levels of Structure}
    \begin{itemize}
      \item Clean, extensively customizable visual style
      
      \item No weird scaling like prosper
      \begin{itemize}
        \item slides are 96~mm~$\times$~128~mm
        
        \item text is 10-12pt on slide
        
        \item slide itself magnified with Adobe Reader/xpdf/gv to fill screen
      \end{itemize}
      
      \item pgf graphics framework easy to use
      
      \item include external JPEG/PNG/PDF figures
      
      \item output directly to pdf: no PostScript hurdles
      
      \item detailed User's Manual (with good presentation advice, too)
    \end{itemize}
  \end{block}
\end{frame}

\subsection{Examples}

\begin{frame}
  \frametitle{Equations, Equations, Equations!}
  
  One classy equation:
  \begin{center}
  \begin{pgfpicture}{-5cm}{0in}{5cm}{2cm}
    \pgfputat{\pgfxy(0,1.5)}{\pgfbox[center,center]{$\displaystyle |\alert<1>{G}| = |\alert<2>{Z(G)}| + \sum_{i = 1}^r \alert<3>{|G : C_G(g_i)|}$}}%
    %
    \pgfsetendarrow{\pgfarrowto}
    \pgfsetlinewidth{0.6pt}
    \only<1>{\alert{
    \pgfputat{\pgfxy(-2.5,.25)}{\pgfbox[center, base]{whole group $G$}}
    \pgfline{\pgfpartway{0.2}{\pgfxy(-2.5, .25)}{\pgfxy(-2,1.5)}}%
            {\pgfpartway{0.8}{\pgfxy(-2.5, .25)}{\pgfxy(-2,1.5)}}%
    }}%
    \only<2>{\alert{
    \pgfputat{\pgfxy(-1,.25)}{\pgfbox[center, base]{center of $G$}}
    \pgfline{\pgfpartway{0.2}{\pgfxy(-1,.25)}{\pgfxy(-1,1.5)}}%
            {\pgfpartway{0.8}{\pgfxy(-1,.25)}{\pgfxy(-1,1.5)}}%
    }}%
    \only<3>{\alert{
    \pgfputat{\pgfxy(2,.25)}{\pgfbox[center, base]{size of $i$th conj.\ class}}
    \pgfline{\pgfpartway{0.23}{\pgfxy(1.4,.25)}{\pgfxy(1,1.5)}}%
            {\pgfpartway{0.8}{\pgfxy(1.4,.25)}{\pgfxy(1,1.5)}}%
    }}%
  \end{pgfpicture}
  \end{center}
  \vspace{-3mm}
  Another equation (Cauchy Derivative Formulas):
  \[
    f^{(n)}(\alert<4>{z}) = \frac{n!}{2\pi i}\int_\Gamma \frac{f(\alert<5>{\zeta})}{(\alert<5>{\zeta} - \alert<4>{z})^{n + 1}} \, d\alert<5>{\zeta},
    \qquad (n = 1, 2, 3, \dotsc)
  \]
  
\end{frame}

\begin{frame}
  \frametitle{Pictures (with graphicx)}
  \begin{columns}
  \begin{column}{4cm}
  \begin{center}
    % replace 'placeholder' with your own graphic
    \includegraphics[width=3.25cm]{3Women.jpg}\\
    Exhibit A: Molinder-zilla
  \end{center}
  \end{column}
  \begin{column}{6cm}
  \begin{center}
    % replace 'placeholder' with your own graphic
    \includegraphics[width=5cm]{Tom+Stew.jpg}\\
    Exhibit B: Tokyo
  \end{center}
  \end{column}
  \end{columns}
\end{frame}

\section{Examples from Atlanta Presentation}

\subsection{Representation Theory of $S_n$}

\begin{frame}
  \frametitle{Bratteli Diagram for $1 < S_2 < S_3$}
  
  \vspace{-5mm}
  \begin{center}
    \only<1>{Graded diagram of proper partitions for $1 < S_2 < \dotsb < S_n$\\\ }
    \only<2>{Each block for $S_n$ \alert{restricts} to a multiplicity-free direct sum of blocks for $S_{n - 1}$ by \alert{removing} a box}
    \only<3-4>{Each pathway through the diagram corresponds to a filled-in partition and a row/column in matrix}
    \only<5>{A pair of paths ending at same diagram specifies a 1-D Fourier space\\\ }
  \end{center}
  \vspace{-5mm}
  \begin{columns}
  \begin{column}{2.5cm}
    \[
      \begin{pmatrix} \alert<3>{\bullet} & & \only<5>{\raisebox{1em}{\makebox[0pt][c]{\smash[t]{\color{blue}$\downarrow$}}}} \\ \only<5>{\makebox[0pt][r]{\color{red}$\to$~~~}} & \bullet & \alert<5>{\bullet} \\ & \alert<4>{\bullet} & \alert<4>{\bullet} \\ & & & \bullet \end{pmatrix}
    \]
    \begin{center}
      $\mathbb{C} S_3$
    \end{center}
  \end{column}
  \begin{column}{8cm}
  \begin{pgfpicture}{-3.5cm}{0cm}{4cm}{4.5cm}
    
    % customized letters for diagram
    \renewcommand{\irrIione}{\action<3-4>{1}}
    
    \renewcommand{\irrIIione}{\action<3>{1}}
    \renewcommand{\irrIIitwo}{\action<3| alert@3>{2}}
    
    \renewcommand{\irrIIiione}{\action<4>{1}}
    \renewcommand{\irrIIiitwo}{\action<4| alert@4>{2}}
    
    \renewcommand{\irrIIIione}{\action<3>{1}}
    \renewcommand{\irrIIIitwo}{\action<3>{2}}
    \renewcommand{\irrIIIithree}{\action<3| alert@3>{3}}
    
    \renewcommand{\irrIIIiione}{\action<4>{1}}
    \renewcommand{\irrIIIiitwo}{\action<4| alert@4>{3}}
    \renewcommand{\irrIIIiithree}{\action<4>{2}}
    
    \renewcommand{\irrIIIiiione}{\,}
    \renewcommand{\irrIIIiiitwo}{\,}
    \renewcommand{\irrIIIiiithree}{\,}
        
    \pgfnodebox{Ii}[virtual]{\pgfxy(0,0)}{\irrIi}{2pt}{2pt}
    \pgfnodebox{IIi}[virtual]{\pgfxy(-1.25,1.5)}{\alert<2>\irrIIi}{2pt}{2pt}
    \pgfnodebox{IIii}[virtual]{\pgfxy(1.25,1.5)}{\alert<2>\irrIIii}{2pt}{2pt}
    \pgfnodebox{IIIi}[virtual]{\pgfxy(-3,3.5)}{\irrIIIi}{2pt}{2pt}
    \pgfnodebox{IIIii}[virtual]{\pgfxy(0,3.5)}{\alert<2>\irrIIIii}{2pt}{2pt}
    \pgfnodebox{IIIiii}[virtual]{\pgfxy(3,3.5)}{\irrIIIiii}{2pt}{2pt}
    
    \pgfnodesetsepstart{0pt}
    \pgfnodesetsepend{0pt}
    \pgfsetlinewidth{0.75pt}
    
    %\only<2-4>{%
    %  \pgfsetendarrow{\pgfarrowpointed}
    %  \pgfnodesetsepend{5pt}}
    {
      \only<3,5>{\pgfsetendarrow{\pgfarrowpointed}\pgfnodesetsepend{5pt}}
      \alert<3,5>{\pgfnodeconnline{Ii}{IIi}}}
    {
      \only<4,5>{\pgfsetendarrow{\pgfarrowpointed}\pgfnodesetsepend{5pt}}
      \alert<4>{\only<5>{\color{blue}}\pgfnodeconnline{Ii}{IIii}}}
    {
      \only<3>{\pgfsetendarrow{\pgfarrowpointed}\pgfnodesetsepend{5pt}}
      \alert<3>{\pgfnodeconnline{IIi}{IIIi}}}
    {
      \only<2>{\pgfsetstartarrow{\pgfarrowpointed}\pgfnodesetsepstart{5pt}}
      \only<5>{\pgfsetendarrow{\pgfarrowpointed}\pgfnodesetsepend{5pt}}
      \alert<2,5>{\pgfnodeconnline{IIi}{IIIii}}}
    {
      \only<2>{\pgfsetstartarrow{\pgfarrowpointed}\pgfnodesetsepstart{5pt}}
      \only<4,5>{\pgfsetendarrow{\pgfarrowpointed}\pgfnodesetsepend{5pt}}
      \alert<2,4>{\only<5>{\color{blue}}\pgfnodeconnline{IIii}{IIIii}}}
    \pgfnodeconnline{IIii}{IIIiii}
    
  \end{pgfpicture}
  \end{column}
  \end{columns}

\end{frame}

\subsection{Construction of Factorization}

\begin{frame}
  \frametitle{Construction of Factorization}
  \begin{columns}
  \begin{column}[T]{6cm}
  \begin{block}{Stages of Subspace Projections}
    \begin{itemize}
      \item Partial paths give $\mathbb{C} S_n$ subspaces
    
      \item At stage for $S_k$, project onto these subspaces
      
      %\item Projections from $S_{k - 1}$-spaces to $S_k$-spaces effectively a small Fourier transform
      
      \item Build sparse factor from projections
      
      \item Full paths by stage for $S_n$
      
      %\item Multiplicity-free restrictions guarantee that each pair of paths corresponds to a 1-D subspace of $\mathbb{C} S_n$
    
    \end{itemize}
  \end{block}
  \end{column}
  \begin{column}[T]{6cm}
    \Yboxdim{11pt}
    \begin{pgfpicture}{-3cm}{0cm}{3cm}{3.5cm}
    \pgfsetxvec{\pgfpoint{0.7cm}{0cm}}
    \pgfsetyvec{\pgfpoint{0cm}{0.8cm}}
    
    % customized letters for diagram
    \renewcommand{\irrIione}{\,}
    
    \renewcommand{\irrIIione}{\,}
    \renewcommand{\irrIIitwo}{\,}
    
    \renewcommand{\irrIIiione}{\,}
    \renewcommand{\irrIIiitwo}{\,}
    
    \renewcommand{\irrIIIione}{\,}
    \renewcommand{\irrIIIitwo}{\,}
    \renewcommand{\irrIIIithree}{\,}
    
    \renewcommand{\irrIIIiione}{\,}
    \renewcommand{\irrIIIiitwo}{\,}
    \renewcommand{\irrIIIiithree}{\,}
    
    \renewcommand{\irrIIIiiione}{\,}
    \renewcommand{\irrIIIiiitwo}{\,}
    \renewcommand{\irrIIIiiithree}{\,}
        
    \pgfnodebox{Ii}[virtual]{\pgfxy(0,0)}{\alert<1-2>\irrIi}{2pt}{2pt}
    \pgfnodebox{IIi}[virtual]{\pgfxy(-1.25,1.5)}{\alert<1>\irrIIi}{2pt}{2pt}
    \pgfnodebox{IIii}[virtual]{\pgfxy(1.25,1.5)}{\alert<2>\irrIIii}{2pt}{2pt}
    \pgfnodebox{IIIi}[virtual]{\pgfxy(-3,3.5)}{\irrIIIi}{2pt}{2pt}
    \pgfnodebox{IIIii}[virtual]{\pgfxy(0,3.5)}{\irrIIIii}{2pt}{2pt}
    \pgfnodebox{IIIiii}[virtual]{\pgfxy(3,3.5)}{\irrIIIiii}{2pt}{2pt}
    
    \pgfnodesetsepstart{0pt}
    \pgfnodesetsepend{0pt}
    \pgfsetlinewidth{0.75pt}
    
    {
      \only<1>{\pgfsetendarrow{\pgfarrowpointed}\pgfnodesetsepend{3pt}}
      \alert<1>{\pgfnodeconnline{Ii}{IIi}}
    }
    {
      \only<2>{\pgfsetendarrow{\pgfarrowpointed}\pgfnodesetsepend{2pt}}
      \alert<2>{\pgfnodeconnline{Ii}{IIii}}
    }
    \pgfnodeconnline{IIi}{IIIi}
    \pgfnodeconnline{IIi}{IIIii}
    \pgfnodeconnline{IIii}{IIIii}
    \pgfnodeconnline{IIii}{IIIiii}
    
  \end{pgfpicture}
  \vspace{2mm}
  \[
    \makebox[0pt][r]{$\mathbb{C} S_3 \cong \;$}\begin{pmatrix} \,\alert<1>\bullet\, \end{pmatrix} \oplus \begin{pmatrix} \,\alert<1>\bullet & \alert<1>\bullet\, \\ \,\alert<2>\bullet & \alert<2>\bullet\, \end{pmatrix} \oplus \begin{pmatrix} \,\alert<2>\bullet\, \end{pmatrix}
%    \begin{pmatrix} \alert<1>\bullet \\ & \alert<1>\bullet & \alert<1>\bullet \\ & \alert<2>\bullet & \alert<2>\bullet \\ & & & \alert<2>\bullet \end{pmatrix}
  \]
  \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{References}
  \footnotesize
  % replace with your own bibtex info
  %adkins-weintraub:1992,
  %\nocite{sagan:1991,maslen:1998}
  %\bibliographystyle{plain}
  %\bibliography{../../reports/annotbib/thesis-biblio}
  
  \begin{block}{On The Web}
    Senior thesis website: $\langle${http://www.math.hmc.edu/$\sim$emalm/thesis/}$\rangle$
  \end{block}
  
  \begin{block}{Beamer Website}
  $\langle${https://sourceforge.net/projects/latex-beamer/}$\rangle$
  \end{block}
\end{frame}

\end{document}
