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\usenavigationsymbolstemplate{}
\mode<presentation>
{
  \useinnertheme{circles}

  \colorlet{themecolor}{blue!90}
  \colorlet{alert}{blue!90}
  \usecolortheme[named= themecolor]{structure}


 
  \setbeamercolor{alerted text}{fg= alert}
  \setbeamercolor{example text}{fg= themecolor}
%  \setbeamercolor{bibliography entry author}{fg=black!90!white}
  %\setbeamercolor{bibliography entry title}{use=normal text,fg=normal text.fg}
  %\setbeamercolor{bibliography entry location}{use=structure,fg=structure.fg!65!bg}
  %\setbeamercolor{bibliography entry note}{use=structure,fg=structure.fg!65!bg}

  \setcounter{secnumdepth}{-1}
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\usefonttheme[%
%  	stillsansserifsmall, 	% will cause 'small' text to be still typeset using sans serif. This refers to
  					% the text in the headline, footline, and sidebars. Using this options is often
  					% advisable since small  text is often easier to read in sans serif.
   	onlymath, % typset math in serif but nothing else
 ]{serif}

\def\RR{\mathbb{R}}


\usepackage{tikz}
\usepackage{graphicx}
\usepackage{epstopdf}
\tikzstyle{V}=[draw, blue, fill=blue, circle, inner sep=0pt, inner sep=0pt, minimum size=2pt]

\def\ShiftX{\pgftransformxshift}
\def\ShiftY{\pgftransformyshift}


\newcommand\Mat[9]{
\node at (1,1) {\small #1} ;
\node at (2,1) {\small#2} ;
\node at (3,1) {\small#3} ;
\node at (1,2) {\small#4} ;
\node at (2,2) {\small#5} ;
\node at (3,2) {\small#6} ;
\node at (1,3) {\small#7} ;
\node at (2,3) {\small#8} ;
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\draw[thick, rounded corners] (3.2,.3)--(3.7, .3)--(3.7,3.7)--(3.2,3.7); 
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\newcommand\MatB[9]{
\draw[thick, |->] (-6,1.5) -- (0,1.5) node [above,align=center,midway]{\scriptsize $A$};
\draw[thick, |->] (-6,3) -- (2,3) node [below,align=center,midway]{\scriptsize $B$};
\draw[fill=blue!10, draw=blue!50, rounded corners = 2pt] (.5,2)--(.5,.5)--(2.5,.5)--(2.5,2.5)--(.5,2.5)--(.5,2);
\draw[fill=yellow!20, draw=orange!50, rounded corners = 2pt] (2.5,3)--(2.5,2.5)--(3.5,2.5)--(3.5,3.5)--(2.5,3.5)--(2.5,3);
\node at (1,1) {\small #1} ;
\node at (2,1) {\small#2} ;
\node at (1,2) {\small#4} ;
\node at (2,2) {\small#5} ;
\node at (3,3) {\small#9} ;
\draw[thick, rounded corners = 2pt] (.8,.5)--(.5, .5)--(.5,2.5)--(.8,2.5); 
\draw[thick, rounded corners = 2pt] (2.2,.5)--(2.5, .5)--(2.5,2.5)--(2.2,2.5); 
\draw[thick, rounded corners = 2pt] (2.8,2.5)--(2.5, 2.5)--(2.5,3.5)--(2.8,3.5); 
\draw[thick, rounded corners = 2pt] (3.2,2.5)--(3.5, 2.5)--(3.5,3.5)--(3.2,3.5); 
}

\newcommand\MatC[9]{\ShiftX{10.25cm}
\draw[thick, |->] (-6,1.5) -- (-1.5,1.5) node [above,align=center,midway]{\scriptsize $A$};
\draw[thick, |->] (-6,3) -- (2,3) node [below,align=center,midway]{\scriptsize $B$};
\draw[fill=blue!10, draw=blue!50, rounded corners = 2pt] (-1,2)--(-1,.5)--(2.5,.5)--(2.5,2.5)--(-1,2.5)--(-1,2);
\draw[fill=yellow!20, draw=orange!50, rounded corners = 2pt] (2.5,3)--(2.5,2.5)--(3.5,2.5)--(3.5,3.5)--(2.5,3.5)--(2.5,3);
\node at (-.125,1) {\tiny #1} ;
\node at (1.625,1) {\tiny #2} ;
\node at (-.125,2) {\tiny #4} ;
\node at (1.625,2) {\tiny#5} ;
\node at (3,3) {\small#9} ;
\draw[thick, rounded corners = 2pt] (-.7,.5)--(-1, .5)--(-1,2.5)--(-.7,2.5); 
\draw[thick, rounded corners = 2pt] (2.2,.5)--(2.5, .5)--(2.5,2.5)--(2.2,2.5); 
\draw[thick, rounded corners = 2pt] (2.8,2.5)--(2.5, 2.5)--(2.5,3.5)--(2.8,3.5); 
\draw[thick, rounded corners = 2pt] (3.2,2.5)--(3.5, 2.5)--(3.5,3.5)--(3.2,3.5); 
}

\def\Blocks{
	\uncover<3>{
		\draw[fill=blue!10, draw=blue!50, rounded corners = 2pt] (.5,2)--(.5,.5)--(2.5,.5)--(2.5,2.5)--(.5,2.5)--(.5,2);
		\draw[fill=yellow!20, draw=orange!50, rounded corners = 2pt] (2.5,3)--(2.5,2.5)--(3.5,2.5)--(3.5,3.5)--(2.5,3.5)--(2.5,3);}}

\newcommand\Perm[3]{
\coordinate (center) at (0,2);
\begin{scope}[scale=1.2, shift=(center)]
\foreach \x in {1,2,3}{
\node[V] (b\x) at (\x,.75) {};
\node[V](t\x) at (\x,-.75) {};
\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b#1);
\draw[->] (t2) -- (b#2);
\draw[->] (t3) -- (b#3);
\end{scope}
}


%For partitions. Takes in a list of row lengths. 
%If you want English convention, you'll need to set yscale=-... as below. 
\newcommand{\Part}[1]{
 \foreach \x [count=\s from 1] in {#1}{
 	{\ifnum\s=1
		\draw (0,\s-1)--(\x,\s-1); 
		\fi}
   \draw (0,\s) to (\x,\s);
   \foreach \y in {0, ..., \x} {\draw (\y,\s)--(\y,\s-1);}
 }}


%For colored partitions. Takes in a list of row lengths and a color. 
 \newcommand{\fPart}[2]{
 \foreach \x [count=\s from 1] in {#1}{
  \filldraw[#2] (0,\s) -- (\x,\s)--(\x,\s-1)--(0,\s-1);
 }
 \foreach \x [count=\s from 1] in {#1}{
 	{\ifnum\s=1
		\draw (0,\s-1)--(\x,\s-1); 
		\fi}
   \draw (0,\s) to (\x,\s);
   \foreach \y in {0, ..., \x} {\draw (\y,\s)--(\y,\s-1);}
 }}

\def\UNIT{.25}
\newcommand{\PART}[1]{
\begin{tikzpicture}[xscale=\UNIT, yscale=-\UNIT] 
	\Part{#1}
\end{tikzpicture}
}\newcommand{\gPART}[1]{
\begin{tikzpicture}[xscale=\UNIT, yscale=-\UNIT] 
\begin{scope}[black!20]
	\Part{#1}
\end{scope}
\end{tikzpicture}
}
\newcommand{\fPART}[2]{
\begin{tikzpicture}[xscale=\UNIT, yscale=-\UNIT]
	\fPart{#1}{#2}
\end{tikzpicture}
}


\def\Scale{.37}



\title{A taste of combinatorial representation theory\\
(yum)}
\author{Zajj Daugherty\\Dartmouth College \& ICERM}
\date{March 14, 2013\\
(Happy $\pi$ day!)}



\begin{document}
\begin{frame}

\maketitle

\end{frame}


\begin{frame}{Permutations and the symmetric group}
Permutation diagrams:
$$\begin{tikzpicture}[line width=1pt]
\uncover<3->{
	\foreach \x in {1,...,5}{
	\draw[dotted, themecolor] (\x,0)--(\x,-.5);
	}
}
%%%%%%%%%%%%%
\draw[->, shorten >=2pt] (1,1.5)--(2,0); 
\draw[->, shorten >=2pt] (2,1.5)--(4,0); 
\draw[->, shorten >=2pt] (3,1.5)--(1,0); 
\draw[->, shorten >=2pt] (4,1.5)--(5,0); 
\draw[->, shorten >=2pt] (5,1.5)--(3,0); 
\foreach \x in {1,...,5}{
 \filldraw [themecolor] (\x,1.5) circle (2pt);
 \filldraw [themecolor] (\x,0) circle (2pt);
\node at (\x, 1.8) {\scriptsize \color{themecolor} \x};
\uncover<1-2>{  \node at (\x, -.3) {\scriptsize \color{themecolor} \x};}
}
%%%%%%%%%%%%%
\pgftransformyshift{-2.1cm}
%%%%%%%%%%%%%
\uncover<2->{
	\draw[->, shorten >=2pt] (1,1.5)--(2,0); 
	\draw[->, shorten >=2pt] (2,1.5)--(1,0); 
	\draw[->, shorten >=2pt] (3,1.5)--(4,0); 
	\draw[->, shorten >=2pt] (4,1.5)--(5,0); 
	\draw[->, shorten >=2pt] (5,1.5)--(3,0); 
	\foreach \x in {1,...,5}{
		 \filldraw [themecolor] (\x,1.5) circle (2pt);
		 \filldraw [themecolor] (\x,0) circle (2pt);
		  \node at (\x, -.3) {\scriptsize \color{themecolor} \x};
	 }
}
%%%%%%%%%%%%%
\pgftransformyshift{1cm}
\pgftransformxshift{6cm}
%%%%%%%%%%%%%
\uncover<4->{\node at (0,.75) {$=$};
	\draw[->, shorten >=2pt] (1,1.5)--(1,0); 
	\draw[->, shorten >=2pt] (2,1.5)--(5,0); 
	\draw[->, shorten >=2pt] (3,1.5)--(2,0); 
	\draw[->, shorten >=2pt] (4,1.5)--(3,0); 
	\draw[->, shorten >=2pt] (5,1.5)--(4,0); 
	\foreach \x in {1,...,5}{
		 \filldraw [themecolor] (\x,1.5) circle (2pt);
		 \filldraw [themecolor] (\x,0) circle (2pt);
		  \node at (\x, -.3) {\scriptsize \color{themecolor} \x};
		  \node at (\x, 1.8) {\scriptsize \color{themecolor} \x};
	 }
}
\end{tikzpicture}$$
\uncover<2->{Permutations ``multiply'' by stacking and resolving.}

\medskip

\uncover<5->{The \alert{symmetric group} $S_n$ is the group of permutations of $1, \dots, n$ with multiplication given by stacking and resolving diagrams.}
\end{frame}




\begin{frame}{Some examples:} 

\def\Unit{.5}
$$
\begin{tikzpicture}[scale=\Unit]
\node at (-2,0) {$S_1:$};
\foreach \x in {1}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b1);
\end{tikzpicture}$$


\hrule
\medskip

$$\begin{tikzpicture}[scale=\Unit]
\node at (-2,0) {$S_2:$};
\foreach \x in {1,2}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b1);
\draw[->] (t2) -- (b2);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=\Unit]
\foreach \x in {1,2}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b2);
\draw[->] (t2) -- (b1);
\end{tikzpicture}
$$


\hrule
\medskip

$S_3:$


$$\begin{tikzpicture}[scale=\Unit]
\foreach \x in {1,2,3}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b1);
\draw[->] (t2) -- (b2);
\draw[->] (t3) -- (b3);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=\Unit]
\foreach \x in {1,2,3}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b2);
\draw[->] (t2) -- (b1);
\draw[->] (t3) -- (b3);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=\Unit]
\foreach \x in {1,2,3}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b1);
\draw[->] (t2) -- (b3);
\draw[->] (t3) -- (b2);
\end{tikzpicture}$$
\medskip
$$\begin{tikzpicture}[scale=\Unit]
\foreach \x in {1,2,3}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b2);
\draw[->] (t2) -- (b3);
\draw[->] (t3) -- (b1);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=\Unit]
\foreach \x in {1,2,3}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b3);
\draw[->] (t2) -- (b1);
\draw[->] (t3) -- (b2);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=\Unit]
\foreach \x in {1,2,3}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};
}
\draw[-> ] (t1) -- (b3);
\draw[->] (t2) -- (b2);
\draw[->] (t3) -- (b1);
\end{tikzpicture}$$

\end{frame}


%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}
A \alert{representation} of a group is a map from the group to a set of matrices which ``preserves structure.''
\pause

\smallskip
\hrule
\smallskip

Example: Permutation representation of the symmetric group. 

$$
\begin{tikzpicture}[scale=1]
\foreach \x in {1,2,3}{
	\node[V] (t\x) at (\x,.75) {};
	\node[V](b\x) at (\x,-.75) {};
\uncover<1-3>{	\node[below, outer sep=3pt, inner sep = 0pt] at (b\x) {\tiny \x};
	\node[above, outer sep=3pt, inner sep = 0pt] at (t\x) {\tiny \x};}
}
\draw[-> ] (t1) -- (b2);
\draw[->] (t2) -- (b3);
\draw[->] (t3) -- (b1);
\uncover<4->{
\foreach \x in {1,2,3}{
\node[above] at (t\x) {\small$v_\x$};
\node[below] at (b\x) {\small$v_\x$};
}
\uncover<5->{
	\node[right] at (3.5,0) {$\displaystyle = \begin{pmatrix}  0&0&1\\ 1&0&0\\0&1&0\end{pmatrix}$ };
}
}
\end{tikzpicture}$$
\uncover<3->{Pick a basis for $\RR^3$: \\
\centerline{$\displaystyle v_1 = \hbox{\scriptsize$\begin{pmatrix} 1\\0\\0 \end{pmatrix}$} \qquad  v_2 =  \hbox{\scriptsize$\begin{pmatrix} 0\\1\\0 \end{pmatrix}$} \qquad v_3 =  \hbox{\scriptsize$\begin{pmatrix} 0\\0\\1 \end{pmatrix}$} $}}\uncover<4->{\noindent Map each permutation to the matrix which permutes the basis vectors in the same way.}



\uncover<6->{
\smallskip
\hrule
\smallskip


Aside: we actually have a representation of the \alert{group ring} \\
\centerline{$\displaystyle \RR S_n = \left\{ \sum_{\sigma \in S_n} r_\sigma \sigma ~|~ r_\sigma \in \RR \right\} , \text{ with multiplication like polynomials } $}}
\end{frame}


%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}
\def\YBlock{
\draw[fill=yellow!10, draw=yellow, rounded corners = 2pt] (.5,2)--(.5,.5)--(3.5,.5)--(3.5,3.5)--(.5,3.5)--(.5,2);
}
\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{1}{2}{3}
\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
	\Mat
	{1}{0}{0}
	{0}{1}{0}
	{0}{0}{1}
\uncover<4->{	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
	\uncover<8->{\YBlock}
	\Mat
	{1}{0}{0}
	{0}{1}{0}
	{0}{0}{1}}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{2}{3}{1}
	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\Mat
	{0}{0}{1}
	{1}{0}{0}
	{0}{1}{0}
\uncover<5->{	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
	\uncover<8->{\YBlock}
\Mat
	{0}{-1}{0}
	{1}{-1}{0}
	{0}{0}{1}}
\end{tikzpicture}}

\medskip

\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{2}{1}{3}
\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\Mat
	{0}{1}{0}
	{1}{0}{0}
	{0}{0}{1}
\uncover<6->{
	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
	\uncover<8->{\YBlock}
\Mat
	{-1}{1}{0}
	{0}{1}{0}
	{0}{0}{1}}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{3}{1}{2}
\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\Mat
	{0}{1}{0}
	{0}{0}{1}
	{1}{0}{0}
\uncover<7->{	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
	\uncover<8->{\YBlock}
\Mat
	{-1}{1}{0}
	{-1}{0}{0}
	{0}{0}{1}}
\end{tikzpicture}}

\medskip

\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{1}{3}{2}
\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\Mat
	{1}{0}{0}
	{0}{0}{1}
	{0}{1}{0}
\uncover<7->{		\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
	\uncover<8->{\YBlock}
\Mat
	{1}{0}{0}
	{1}{-1}{0}
	{0}{0}{1}}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{3}{2}{1}
	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\Mat
	{0}{0}{1}
	{0}{1}{0}
	{1}{0}{0}
\uncover<7->{		\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
	\uncover<8->{\YBlock}
\Mat
	{0}{-1}{0}
	{-1}{0}{0}
	{0}{0}{1}}
\end{tikzpicture}}

\uncover<2->{
Notice that the permutation representation has an \alert{invariant subspace} $\RR\{v_1 + v_2 + v_3\}$, since 
$$M (v_1 + v_2 + v_3) = v_1 + v_2 + v_3 $$
for all permutation matrices $M$.}

\uncover<3->{Change to basis\\
$$w_1 = v_1 - v_2, \qquad w_2 = v_2 - v_3, \qquad \alert{w_3 = v_1 + v_2 + v_3}$$}

\end{frame}



%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}
\def\GBlock{
\draw[fill=green!10, draw=green!50!blue, rounded corners = 2pt] (.5,2)--(.5,.5)--(3.5,.5)--(3.5,3.5)--(.5,3.5)--(.5,2);
}
\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{1}{2}{3}
\uncover<1-3>{	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\GBlock
	\Mat
	{1}{0}{0}
	{0}{1}{0}
	{0}{0}{1}
\uncover<2-3>{	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
	\Blocks
	\Mat
	{1}{0}{0}
	{0}{1}{0}
	{0}{0}{1}}}
\uncover<4-6>{
	\MatB
	{1}{0}{0}
	{0}{1}{0}
	{0}{0}{1}}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{2}{3}{1}
\uncover<1-3>{	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\GBlock
\Mat
	{0}{0}{1}
	{1}{0}{0}
	{0}{1}{0}}
\uncover<2-3>{	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
\Blocks
\Mat
	{0}{-1}{0}
	{1}{-1}{0}
	{0}{0}{1}}
\uncover<4-6>{
\MatB
	{0}{-1}{0}
	{1}{-1}{0}
	{0}{0}{1}}
\end{tikzpicture}}

\medskip

\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{2}{1}{3}
\uncover<1-3>{	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\GBlock
\Mat
	{0}{1}{0}
	{1}{0}{0}
	{0}{0}{1}
\uncover<2-3>{	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
\Blocks
\Mat
	{-1}{1}{0}
	{0}{1}{0}
	{0}{0}{1}}}
\uncover<4-6>{
\MatB
	{-1}{1}{0}
	{0}{1}{0}
	{0}{0}{1}}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{3}{1}{2}
\uncover<1-3>{	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\GBlock
\Mat
	{0}{1}{0}
	{0}{0}{1}
	{1}{0}{0}}
\uncover<2-3>{	\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
\Blocks
\Mat
	{-1}{1}{0}
	{-1}{0}{0}
	{0}{0}{1}}
\uncover<4-6>{
\MatB
	{-1}{1}{0}
	{-1}{0}{0}
	{0}{0}{1}}
\end{tikzpicture}}

\medskip

\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{1}{3}{2}
\uncover<1-3>{	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\GBlock
\Mat
	{1}{0}{0}
	{0}{0}{1}
	{0}{1}{0}}
\uncover<2-3>{		\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
\Blocks
\Mat
	{1}{0}{0}
	{1}{-1}{0}
	{0}{0}{1}}
\uncover<4-6>{
\MatB
	{1}{0}{0}
	{1}{-1}{0}
	{0}{0}{1}}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{3}{2}{1}
\uncover<1-3>{	\ShiftX{3.75cm}\node at (1,2) {$\mapsto$};\ShiftX{1.5cm}
\GBlock
\Mat
	{0}{0}{1}
	{0}{1}{0}
	{1}{0}{0}}
\uncover<2-3>{		\ShiftX{3.5cm}\node at (1,2) {$\sim$};\ShiftX{1.5cm}	
\Blocks
\Mat
	{0}{-1}{0}
	{-1}{0}{0}
	{0}{0}{1}}
\uncover<4-6>{
\MatB
	{0}{-1}{0}
	{-1}{0}{0}
	{0}{0}{1}
}
\end{tikzpicture}}

Start with the {\color{green!70!blue}permutation representation $P$} with basis $\{v_1, v_2, v_3\}$.\\
\uncover<2->{Change to basis
$$w_1 = v_1 - v_2, \qquad w_2 = v_2 - v_3, \qquad w_3 = v_1 + v_2 + v_3$$}\uncover<5->{We say 
{\color{green!70!blue}$P$} is isomorphic to the sum of two smaller representations: 
\centerline{${\color{green!70!blue}P} \cong {\color{blue}A} \oplus {\color{orange!90!yellow}B}$}}
\uncover<6->{We say $A$ and $B$ are \alert{simple} because neither has any invariant subspaces.}
\end{frame}

%%%%%%%%%%%
%%%%%%%%%%%
%%%%%%%%%%%
\begin{frame}
\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{1}{2}{3}
\MatC
	{1}{0}{}
	{0}{1}{}
	{}{}{1}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{2}{3}{1}
\MatC
	{-$1/2$}{-$\sqrt{3}/2$}{}
	{$\sqrt{3}/2$}{-$1/2$}{}
	{}{}{1}
\end{tikzpicture}}

\medskip

\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{2}{1}{3}
\MatC
	{-1}{0}{}
	{0}{1}{}
	{}{}{1}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{3}{1}{2}
\MatC
	{-$1/2$}{$\sqrt{3}/2$}{}
	{-$\sqrt{3}/2$}{-$1/2$}{}
	{}{}{1}
\end{tikzpicture}}

\medskip

\centerline{\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{1}{3}{2}
\MatC
	{$1/2$}{$\sqrt{3}/2$}{}
	{$\sqrt{3}/2$}{-$1/2$}{}
	{}{}{1}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\Perm{3}{2}{1}
\MatC
	{-$1/2$}{$\sqrt{3}/2$}{}
	{$\sqrt{3}/2$}{$1/2$}{}
	{}{}{1}
\end{tikzpicture}}


Start with the {\color{green!70!blue}permutation representation $P$} with basis $\{v_1, v_2, v_3\}$.\\
Change to basis
$${\color{red}w_1 = \sqrt{3}(v_1 - v_2)}, \qquad {\color{red}w_2 = v_1 + v_2 - 2 v_3}, \qquad w_3 = v_1 + v_2 + v_3$$
We say 
{\color{green!70!blue}$P$} is isomorphic to the sum of two smaller representations: \\
\centerline{${\color{green!70!blue}P} \cong {\color{blue}A} \oplus {\color{orange!90!yellow}B}$}
We say $A$ and $B$ are \alert{simple} because neither has any invariant subspaces.
\end{frame}


%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\begin{frame}{How about some combinatorics?}

Let $n$ be a non-negative integer. \\
A \alert{partition $\lambda$} of $n$ is a non-ordered list of positive integers which sum to $n$. 

\bigskip 

Example: the partitions of 3 are $(3)$, $(2,1)$, and $(1,1,1)$. 

\bigskip \pause

We draw partitions as $n$ boxes piled up and to the left, where the \alert{parts} are the number of boxes in a row:

$$\begin{tikzpicture}[xscale=.5, yscale=-.5]
\node[left] at (-.5,2) {$\lambda = (5,4,4,2) = $};
\Part{5,4,4,2}
\end{tikzpicture}$$




\end{frame}




%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\begin{frame}


{
\def\One{2}
\def\Two{3}
\def\TwoA{4}
\def\TwoB{5}
\def\Three{6}
\def\Four{7}
\def\DefA{8}
\def\ExAa{9}
\def\ExAb{10}
\def\ExAc{11}
\def\ExAd{12}
\def\RepsA{13}
\def\RepsB{14}


\begin{center}
\begin{tikzpicture}[xscale=3*\UNIT, yscale=-4.5*\UNIT]
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (0) at (0,.4);
	\coordinate (11) at (0,1.2);
	\coordinate (21) at (-1,2);\coordinate (22) at (1,2);
	\foreach \x in {1, ..., 3}{\coordinate (3\x) at (-4+2*\x,3);}
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-6+2*\x,4.5);}
%edges in lattice:
\uncover<\One->{\draw (0)--(11);}
\uncover<\Two->{
	\draw (11)--(21) (11)--(22);}
	\uncover<\TwoB->{
	\draw (21)--(31) (21)--(32);}
	\uncover<\Three->{\draw (22)--(32) (22)--(33);}
\uncover<\Four->{	
	\draw (31)--(41) (31)--(42) (32)--(42)  (32)--(43) (32)--(44) (33)--(44)  (33)--(45);}
\uncover<\ExAb>{ \draw[thick, red] (0)--(11)--(21)--(31)--(42);}
\uncover<\ExAc>{ \draw[thick, red] (0)--(11)--(21)--(32)--(42);}
\uncover<\ExAd>{ \draw[thick, red] (0)--(11)--(22)--(32)--(42);}
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (0) {$\emptyset$};
\uncover<\One->{	
	\node at (11) {\PART{1}};}
\uncover<\Two->{
	\node at (22) {\PART{1,1}};
	\node at (21) {\PART{2}};
	\only<\TwoA-\TwoB>{\node at (21) {\fPART{2}{red!30}};}
	}
\uncover<\Three->{
	\node at (33) {\PART{1,1,1}};}
\uncover<\TwoB->{	
	\node at (32) {\PART{2,1}};
	\node at (31) {\PART{3}};}
\uncover<\Four->{
	\node at (45) {\PART{1,1,1,1}};
	\node at (44) {\PART{2,1,1}};
	\node at (43) {\PART{2,2}};
	\node at (42) {\PART{3,1}};
	\node at (41) {\PART{4}};}
\uncover<\ExAa-\ExAd>{
	\node at (42) {\fPART{3,1}{red!30}};
}
\uncover<\ExAb>{
	\node at (11) {\fPART{1}{red!30}};
	\node at (21) {\fPART{2}{red!30}};
	\node at (31) {\fPART{3}{red!30}};}
\uncover<\ExAc>{
	\node at (11) {\fPART{1}{red!30}};
	\node at (21) {\fPART{2}{red!30}};
	\node at (32) {\fPART{2,1}{red!30}};}
\uncover<\ExAd>{
	\node at (11) {\fPART{1}{red!30}};
	\node at (22) {\fPART{1,1}{red!30}};
	\node at (32) {\fPART{2,1}{red!30}};}
\uncover<\RepsA->{
	\node at (31) {\fPART{3}{yellow!30}};
	\node at (32) {\fPART{2,1}{blue!30}};
	\node at (33) {\fPART{1,1,1}{red!30}};}
\end{scope}
\uncover<\RepsA->{\node at (-4,3) {$S_3:$};}
\uncover<\Four->{\foreach \x in {-4,0,4} {\node at (\x , 5.1) {$\vdots$}; }}
\node[below right] at (-6,.4) {\alert{Young's lattice:}};
\end{tikzpicture}
\end{center}


\uncover<\DefA->{\alert{$\lambda$-Tableau}: a path from $\emptyset$ down to a partition $\lambda$. \\}
\uncover<\RepsA->{\alert{Theorem 1}: (Up to isomorphism) the simple $S_n$-representations are \alert{indexed} by partitions of $n$.\\}
\uncover<\RepsB->{\alert{Theorem 2}: If $\lambda$ is a partition of $n$, then the corresponding representation has basis indexed by $\lambda$-tableaux, and matrices determined by other combinatorial data about those paths.}


}\end{frame}


%%%%%%%%%%%%
%%%%%%%%%%%%
%%%%%%%%%%%%
%%%%%%%%%%%%
\begin{frame}

\begin{center}
\begin{tikzpicture}[xscale=3*\UNIT, yscale=-4.5*\UNIT]
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (00) at (-2,.4);
	\coordinate (01) at (0,.4);
	\coordinate (02) at (2,.4);
	\foreach \x in {0, ..., 2}{\coordinate (1\x) at (-2+2*\x,1.2);}
	\foreach \x in {0, ..., 3}{\coordinate (2\x) at (-3+2*\x,2);}
	\foreach \x in {0, ..., 4}{\coordinate (3\x) at (-4+2*\x,3);}
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-6+2*\x,4.5);}
	\draw[very thin, black!20] (21)--(31) (22)--(33) (31)--(41) (31)--(42) (32)--(42)  (32)--(43) (32)--(44) (33)--(44)  (33)--(45);
%edges in lattice:
	\draw[thick, blue] (01)--(11)--(21) (11)--(22) (21)--(32) (22)--(32);
	\draw[thick, orange] (00)--(10)--(20)--(30);
	\draw[thick, red] (02)--(12)--(23)--(34);
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (33) {\gPART{1,1,1}};
	\node at (31) {\gPART{3}};
	\node at (45) {\gPART{1,1,1,1}};
	\node at (44) {\gPART{2,1,1}};
	\node at (43) {\gPART{2,2}};
	\node at (42) {\gPART{3,1}};
	\node at (41) {\gPART{4}};
\end{scope}
\begin{scope}[every node/.style={fill=white}]
	\node at (00) {\color{orange}$\emptyset$};
	\node at (01) {\color{blue}$\emptyset$};
	\node at (02) {\color{red}$\emptyset$};
	\node at (10) {\fPART{1}{yellow!30}};
	\node at (20) {\fPART{2}{yellow!30}};
	\node at (30) {\fPART{3}{yellow!30}};
	\node at (11) {\fPART{1}{blue!30}};
	\node at (21) {\fPART{2}{blue!30}};
	\node at (22) {\fPART{1,1}{blue!30}};
	\node at (32) {\fPART{2,1}{blue!30}};
	\node at (12) {\fPART{1}{red!30}};
	\node at (23) {\fPART{1,1}{red!30}};
	\node at (34) {\fPART{1,1,1}{red!30}};
\end{scope}
\foreach \x in {-4,0,4} {\node[white] at (\x , 5.1) {$\vdots$}; }
\end{tikzpicture}
\end{center}


\alert{$\lambda$-Tableau}: a path from $\emptyset$ to a partition $\lambda$. \\
\alert{Theorem 1}: (Up to isomorphism) the simple $S_n$-representations are \alert{indexed} by partitions of $n$.\\
\alert{Theorem 2}: If $\lambda$ is a partition of $n$, then the corresponding representation has basis indexed by $\lambda$-tableaux, and matrices determined by other combinatorial data about those paths.

\end{frame}


%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}

\begin{center}
\begin{tikzpicture}[xscale=3*\UNIT, yscale=-4.5*\UNIT]
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (00) at (-2,.4);
	\coordinate (01) at (0,.4);
	\coordinate (02) at (2,.4);
	\foreach \x in {0, ..., 2}{\coordinate (1\x) at (-2+2*\x,1.2);}
	\foreach \x in {0, ..., 3}{\coordinate (2\x) at (-3+2*\x,2);}
	\foreach \x in {0, ..., 4}{\coordinate (3\x) at (-4+2*\x,3);}
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-6+2*\x,4.5);}
\uncover<3->{
\begin{scope}[very thin, blue]
	\draw(01)--(11) node[midway, left] {\tiny 0};
	\draw(11)--(21) node[midway, above left] {\tiny 1};
	\draw(11)--(22) node[midway, above right ] {\tiny -1};
	\draw(21)--(32) node[midway, below left] {\tiny -1};
	\draw(22)--(32) node[midway, below right] {\tiny 1};
\end{scope}
\begin{scope}[very thin, orange]
	\draw(00)--(10) node[midway, left] {\tiny 0};
	\draw(10)--(20) node[midway, above left] {\tiny 1};
	\draw(20)--(30) node[midway, above left] {\tiny 2};
\end{scope}
\begin{scope}[very thin, red]
	\draw(02)--(12) node[midway, left] {\tiny 0};
	\draw(12)--(23) node[midway, above right ] {\tiny -1};
	\draw(23)--(34) node[midway, above right] {\tiny -2};
\end{scope}}
%edges in lattice:
	\draw[thick, blue] (01)--(11)--(21) (11)--(22) (21)--(32) (22)--(32);
	\draw[thick, orange] (00)--(10)--(20)--(30);
	\draw[thick, red] (02)--(12)--(23)--(34);
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (00) {\color{orange}$\emptyset$};
	\node at (01) {\color{blue}$\emptyset$};
	\node at (02) {\color{red}$\emptyset$};
	\node at (10) {\fPART{1}{yellow!30}};
	\node at (20) {\fPART{2}{yellow!30}};
	\node at (30) {\fPART{3}{yellow!30}};
	\node at (11) {\fPART{1}{blue!30}};
	\node at (21) {\fPART{2}{blue!30}};
	\node at (22) {\fPART{1,1}{blue!30}};
	\node at (32) {\fPART{2,1}{blue!30}};
	\node at (12) {\fPART{1}{red!30}};
	\node at (23) {\fPART{1,1}{red!30}};
	\node at (34) {\fPART{1,1,1}{red!30}};
\end{scope}
\end{tikzpicture}
\end{center}

What other combinatorial data?

\uncover<2->{
The \alert{content} of a box in a partition is its diagonal number:\\
\centerline{\begin{tikzpicture}[xscale=.35, yscale=-.35]
\node[left] at (-1,2) {$\lambda = (5,4,4,2) = $};
\Part{5,4,4,2}
\foreach \y in {1,...,3} {\draw[green!70!blue] (0, \y)--(-.3, \y-.3) node[above left, inner sep=0pt] {\tiny -\y};}
\foreach \x in {0,...,4} {\draw[green!70!blue] (\x,0)--(\x-.3, -.3) node[above left, inner sep=0pt] {\tiny \x};}
\foreach \y/\x  in {1/1,1/2,1/3,1/4,1/5,2/1,2/2,2/3,2/4,3/1,3/2,3/3,3/4,4/1,4/2} {
\pgfmathsetmacro{\d}{\x-\y};
\node[green!70!blue] at (\x-.5, \y-.5) {\tiny \pgfmathprintnumber{\d}};
}
\end{tikzpicture}}}


\uncover<4->{Again:\\
$*$ each partition is secretly a representation\\
$*$ each path is secretly a basis vector\\
Now: entries in matrices for $s_1$, $s_2$, \dots, are given by expressions in the contents of boxes added. }
\end{frame}




%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}
The rule for $s_i$:
\medskip
\pause

\begin{minipage}[t]{4cm}
Suppose {\color{blue}$v$} goes with the path

\smallskip
\centerline{\begin{tikzpicture}[xscale=.5, yscale=-.5]
\begin{scope}[every node/.style={fill=blue, circle, blue, draw, inner sep=0pt, outer sep = 2pt, minimum size=3pt}]
\node(1) at (1,1) {};
\node (2) at (0,2) {};
\node  (3) at (-1,3) {};
\node  (4) at (0,4) {};
\node (5) at (-1,5) {};
\end{scope}
\node[left, blue] at (-1.75,3) {\footnotesize level $i$:};
\draw[thick, blue] (1)--(2)--(3)--(4)--(5);
\draw[thick, dotted, blue] (1.5,.5)--(1) (5)--(-1.5,5.5);
\uncover<3->{
\node[fill, circle, green!80!blue, draw, inner sep=0pt, outer sep = 2pt, minimum size=3pt]  (3a) at (1,3) {};
\draw[thick, green!80!blue] (2)--(3a)--(4);
}
\end{tikzpicture}}
\smallskip

\uncover<3->{and {\color{green!70!blue} $u$} is almost the same, except at the $i$th step. }

~\\~\\~\\~\\~\\

\end{minipage}\qquad \begin{minipage}[t]{6cm}
\uncover<4->{\centerline{\def\ExAa{4}\def\ExAb{5}\def\ExAc{6}\def\ExAd{7}
\begin{tikzpicture}[xscale=3*\UNIT, yscale=-4.5*\UNIT]
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (0) at (0,.4);
	\coordinate (11) at (0,1.2);
	\coordinate (21) at (-1,2);\coordinate (22) at (1,2);
	\foreach \x in {1,2}{\coordinate (3\x) at (-4+2*\x,3);}
	\foreach \x in {2}{\coordinate (4\x) at (-6+2*\x,4.5);}
%edges in lattice:
\draw (0)--(11) node[midway, left] {\tiny 0};
\draw (11)--(21) node[midway, above left] {\tiny 1};
\draw (11)--(22) node[midway, above right] {\tiny -1};
\draw (21)--(31) node[midway, above left] {\tiny 2};
\draw (21)--(32) node[midway, above right] {\tiny -1};
\draw (22)--(32) node[midway, below right] {\tiny 1};	
	\draw (31)--(42) node[midway, left] {\tiny -1};
	\draw (32)--(42)  node[midway, below right] {\tiny 2};
\uncover<\ExAb>{ \draw[thick, red] (0)--(11)--(21)--(31)--(42);}
\uncover<\ExAc>{ \draw[thick, red] (0)--(11)--(21)--(32)--(42);}
\uncover<\ExAd>{ \draw[thick, red] (0)--(11)--(22)--(32)--(42);}
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (0) {$\emptyset$};
	\node at (11) {\PART{1}};
	\node at (22) {\PART{1,1}};
	\node at (21) {\PART{2}};
	\node at (32) {\PART{2,1}};
	\node at (31) {\PART{3}};
	\node at (42) {\PART{3,1}};
\uncover<\ExAa-\ExAd>{
	\node at (42) {\fPART{3,1}{red!30}};
}
\uncover<\ExAb>{
	\node at (11) {\fPART{1}{red!30}};
	\node at (21) {\fPART{2}{red!30}};
	\node at (31) {\fPART{3}{red!30}};}
\uncover<\ExAc>{
	\node at (11) {\fPART{1}{red!30}};
	\node at (21) {\fPART{2}{red!30}};
	\node at (32) {\fPART{2,1}{red!30}};}
\uncover<\ExAd>{
	\node at (11) {\fPART{1}{red!30}};
	\node at (22) {\fPART{1,1}{red!30}};
	\node at (32) {\fPART{2,1}{red!30}};}
\end{scope}
\end{tikzpicture}}}
\end{minipage}
\end{frame}

%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}
The rule for $s_i$:
\medskip

\begin{minipage}[t]{4cm}
Suppose {\color{blue}$v$} goes with the path

\smallskip
\centerline{\begin{tikzpicture}[xscale=.5, yscale=-.5]
\begin{scope}[every node/.style={fill=blue, circle, blue, draw, inner sep=0pt, outer sep = 2pt, minimum size=3pt}]
\node(1) at (1,1) {};
\node (2) at (0,2) {};
\node  (3) at (-1,3) {};
\node  (4) at (0,4) {};
\node (5) at (-1,5) {};
\end{scope}
\uncover<2->{
\begin{scope}[blue]
	\draw (2)--(3) node[midway, above left, inner sep=0pt]{\tiny $c_i$};
	\draw (3)--(4) node[midway, below left, inner sep=0pt]{\tiny $c_{i+1}$};
\end{scope}
}
\node[left, blue] at (-1.75,3) {\footnotesize level $i$:};
\draw[thick, blue] (1)--(2)--(3)--(4)--(5);
\draw[thick, dotted, blue] (1.5,.5)--(1) (5)--(-1.5,5.5);
\node[fill, circle, green!80!blue, draw, inner sep=0pt, outer sep = 2pt, minimum size=3pt]  (3a) at (1,3) {};
	\uncover<3->{
	\begin{scope}[green!80!blue]
		\draw (2)--(3a) node[midway, above right, inner sep=0pt]{\tiny $c_{i+1}$};
		\draw (3a)--(4) node[midway, below right, inner sep=0pt]{\tiny $c_{i}$};
	\end{scope}
	}
\draw[thick, green!80!blue] (2)--(3a)--(4);
\end{tikzpicture}}
\smallskip

{and {\color{green!70!blue} $u$} is almost the same, except at the $i$th step. }

\bigskip


\uncover<5->{Back to $S_3$: \\
\begin{tikzpicture}[xscale=4.9*\UNIT, yscale=-5*\UNIT]
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (0v) at (0,0);	\coordinate (0u) at (0,1);
	\coordinate (1v) at (1,0);	\coordinate (1u) at (1,1);
	\coordinate (2v) at (2,0);	\coordinate (2u) at (2,1);
	\coordinate (3v) at (3,0);	\coordinate (3u) at (3,1);
\node[left] at (-.5,0) {$v$:}; \node[left] at (-.5,1) {$u$:}; 
\begin{scope}[blue]
	\draw(0v)--(1v) node[midway, above] {\tiny 0};
	\draw(1v)--(2v) node[midway, above ] {\tiny -1};
	\draw(2v)--(3v) node[midway, above  ] {\tiny 1};
\end{scope}
\begin{scope}[blue]
	\draw(0u)--(1u) node[midway, above] {\tiny 0};
	\draw(1u)--(2u) node[midway, above ] {\tiny 1};
	\draw(2u)--(3u) node[midway, above  ] {\tiny -1};
\end{scope}
\begin{scope}[every node/.style={fill=white}]
	\node at (0u) {\color{blue}$\emptyset$};	\node at (0v) {\color{blue}$\emptyset$};
	\node at (1u) {\fPART{1}{blue!30}};	\node at (1v) {\fPART{1}{blue!30}};
	\uncover<6-7>{\node at (1u) {\fPART{1}{green!30}};	\node at (1v) {\fPART{1}{green!30}};}
	\node at (2u) {\fPART{2}{blue!30}};	\node at (2v) {\fPART{1,1}{blue!30}};
	\uncover<8->{\node at (2u) {\fPART{2}{green!30}};	\node at (2v) {\fPART{1,1}{green!30}};}
	\node at (3u) {\fPART{2,1}{blue!30}};	\node at (3v) {\fPART{2,1}{blue!30}};
\end{scope}
\end{tikzpicture}}



\end{minipage}\qquad \begin{minipage}[t]{6cm}
{Let $c_i$ be the content of the box added from $i-1$ to $i$.}

\bigskip

\uncover<4->{
Then the coefficient in $s_i \cdot v$ \\
\dots on $v$ is $1/(c_{i+1} - c_i)$\\
\dots on $u$ is $\sqrt{ 1- (1/(c_{i+1} - c_i))^2}$\\
\dots on any other path is 0. }

\bigskip

\uncover<6->{
$$
\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]
\only<1-7>{
	\node[left] at (1,2) {$s_1 = $};
	\Perm{2}{1}{3}}
\only<8->{
	\node[left] at (1,2) {$s_2 = $};
	\Perm{1}{3}{2}}
\end{tikzpicture}
$$
$$\begin{array}{c|c@{}|@{}c}
& v & u\\\hline
v &\uncover<7->{\only<1-7>{1/(-1-0)}}
		\uncover<9->{\only<8->{1/(1-(-1))}}& 
	\uncover<7->{\only<1-7>{0}}
		\uncover<10->{\only<8->{\sqrt{1-1/4}}} \\\hline
u&  
	\uncover<7->{\only<1-7>{0}}
		\uncover<10->{\only<8->{\sqrt{1-1/4}}}  &
	\uncover<7->{\only<1-7>{1/(1-0)}}
		\uncover<9->{\only<8->{1/(-1-1)}}  
\end{array}$$
}
\end{minipage}



\end{frame}

%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}

\begin{center}
\begin{tikzpicture}[xscale=1.5*\UNIT, yscale=-4.5*\UNIT]
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (00) at (-4,.4);
	\coordinate (01) at (0,.4);
	\coordinate (02) at (4,.4);
	\foreach \x in {0, ..., 2}{\coordinate (1\x) at (-4+4*\x,1.2);}
	\coordinate (20) at (-6,2);
	\coordinate (21) at (-2,2);
	\coordinate (22) at (2,2);
	\coordinate (23) at (6,2);
	\foreach \x in {0, ..., 4}{\coordinate (3\x) at (-8+4*\x,3);}
\begin{scope}[very thin, blue]
	\draw(01)--(11) node[midway, left] {\tiny 0};
	\draw(11)--(21) node[midway, above left] {\tiny 1};
	\draw(11)--(22) node[midway, above right ] {\tiny -1};
	\draw(21)--(32) node[midway, below left] {\tiny -1};
	\draw(22)--(32) node[midway, below right] {\tiny 1};
\end{scope}
\begin{scope}[very thin, orange]
	\draw(00)--(10) node[midway, left] {\tiny 0};
	\draw(10)--(20) node[midway, above left] {\tiny 1};
	\draw(20)--(30) node[midway, above left] {\tiny 2};
\end{scope}
\begin{scope}[very thin, red]
	\draw(02)--(12) node[midway, left] {\tiny 0};
	\draw(12)--(23) node[midway, above right ] {\tiny -1};
	\draw(23)--(34) node[midway, above right] {\tiny -2};
\end{scope}
%edges in lattice:
	\draw[thick, blue] (01)--(11)--(21) (11)--(22) (21)--(32) (22)--(32);
	\draw[thick, orange] (00)--(10)--(20)--(30);
	\draw[thick, red] (02)--(12)--(23)--(34);
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (00) {\color{orange}$\emptyset$};
	\node at (01) {\color{blue}$\emptyset$};
	\node at (02) {\color{red}$\emptyset$};
	\node at (10) {\fPART{1}{yellow!30}};
	\node at (20) {\fPART{2}{yellow!30}};
	\node at (30) {\fPART{3}{yellow!30}};
	\node at (11) {\fPART{1}{blue!30}};
	\node at (21) {\fPART{2}{blue!30}};
	\node at (22) {\fPART{1,1}{blue!30}};
	\node at (32) {\fPART{2,1}{blue!30}};
	\node at (12) {\fPART{1}{red!30}};
	\node at (23) {\fPART{1,1}{red!30}};
	\node at (34) {\fPART{1,1,1}{red!30}};
\end{scope}
\node[left] at (-12, 4) {\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]\Perm{2}{1}{3}\end{tikzpicture}};
\node at (-8, 4) {$\begin{pmatrix}1\end{pmatrix}$};
\node at (8, 4) {$\begin{pmatrix}-1\end{pmatrix}$};
\node at (0, 4) {$\begin{pmatrix}-1&0\\0&1\end{pmatrix}$};
\node[left] at (-12, 5.5) {\begin{tikzpicture}[xscale=\Scale, yscale=-\Scale]\Perm{1}{3}{2}\end{tikzpicture}};
\node at (-8, 5.5) {$\begin{pmatrix}1\end{pmatrix}$};
\node at (8, 5.5) {$\begin{pmatrix}-1\end{pmatrix}$};
\node at (0, 5.5) {$\begin{pmatrix}1/2&\sqrt{3}/2\\\sqrt{3}/2&-1\end{pmatrix}$};
\node[orange] at (-8, 6.5) {``trivial''};
\node[red] at (8, 6.5) {``alternating''};
\end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}{Counting tableaux and dimensions}

\centerline{\begin{tikzpicture}[xscale=3*\UNIT, yscale=-4.5*\UNIT]
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (0) at (0,.4);
	\coordinate (11) at (0,1.2);
	\coordinate (21) at (-1,2);\coordinate (22) at (1,2);
	\foreach \x in {1, ..., 3}{\coordinate (3\x) at (-4+2*\x,3);}
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-6+2*\x,4.5);}
%edges in lattice:
	\draw (0)--(11);% node[midway, left] {\tiny 0};
	\draw (11)--(21);% node[midway, above left ] {\tiny 1};
	\draw (11)--(22);% node[midway, above right] {\tiny -1};
	\draw (21)--(31);% node[midway, above left] {\tiny 2};
	\draw (21)--(32);% node[midway, above right] {\tiny -1};
	\draw (22)--(32);% node[midway, above left] {\tiny 1};
	\draw (22)--(33);% node[midway, above right] {\tiny -1};
	\draw (31)--(41);% node[midway, above left] {\tiny 3};
	\draw (31)--(42);% node[midway,left ] {\tiny -1};
	\draw (32)--(42);% node[midway, above left] {\tiny 2};
	\draw (32)--(43);% node[midway, left] {\tiny 0};
	\draw (32)--(44);%  node[midway, above right] {\tiny -2};
	\draw (33)--(44);%   node[midway, right] {\tiny 1};
	\draw (33)--(45);% node[midway, above right] {\tiny -3};
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (0) {$\emptyset$};
	\node at (11) {~ \PART{1} \small \alert{1}};
	\node at (22) {~\PART{1,1}  \small \alert{1}};
	\node at (21) {~\PART{2} \small \alert{1}};
	\node at (33) {~\PART{1,1,1} \small \alert{1}};
	\node at (32) {~\PART{2,1} \small \alert{2}};
	\node at (31) {~\PART{3} \small \alert{1}};
	\node at (45) {~\PART{1,1,1,1} \small \alert{1}};
	\node at (44) {~\PART{2,1,1} \small \alert{3}};
	\node at (43) {~\PART{2,2} \small \alert{2}};
	\node at (42) {~\PART{3,1} \small \alert{3}};
	\node at (41) {~\PART{4} \small \alert{1}};
\end{scope}
\uncover<2>{
\node[right] at (5.25,1.2) {\small $1=\alert{1}$};
\node[right] at (5.25,2) {\small$1+1=\alert{2}$};
\node[right] at (5.25,3) {\small$1+2+1=\alert{4}$};
\node[right] at (5.25,4.5) {\small$1+3+2+3+1=\alert{10}$};
}
\uncover<3>{
\node[right] at (5.25,1.2) {\small $1^2=\alert{1}$};
\node[right] at (5.25,2) {\small$1^2+1^2=\alert{2}$};
\node[right] at (5.25,3) {\small$1^2+2^2+1^2=\alert{6}$};
\node[right] at (5.25,4.5) {\small$1^2+3^2+2^2+3^2+1^2=\alert{24}$};
}
\uncover<4>{
\node[right] at (5.25,1.2) {\small $1^2=\alert{1!}$};
\node[right] at (5.25,2) {\small$1^2+1^2=\alert{2!}$};
\node[right] at (5.25,3) {\small$1^2+2^2+1^2=\alert{3!}$};
\node[right] at (5.25,4.5) {\small$1^2+3^2+2^2+3^2+1^2=\alert{4!}$};
}
\uncover<5->{
\node[right] at (5.25,1.2) {\small $1^2=\alert{|S_1|}$};
\node[right] at (5.25,2) {\small$1^2+1^2=\alert{|S_2|}$};
\node[right] at (5.25,3) {\small$1^2+2^2+1^2=\alert{|S_3|}$};
\node[right] at (5.25,4.5) {\small$1^2+3^2+2^2+3^2+1^2=\alert{|S_4|}$};
}
\end{tikzpicture}}

\uncover<6->{
\alert{Wedderburn's theorem:} ``Nice'' rings are isomorphic to the direct sum of matrix rings. }

\uncover<7->{
$$\RR S_n \cong M_1(\RR) \oplus M_2 (\RR) \oplus M_1(\RR)\cong \fPART{3}{yellow!30} \oplus \fPART{2,1}{blue!30} \oplus \fPART{1,1,1}{red!30}$$
\centerline{\small (The matrix ring on an $m$-dimensional v.s.\ is $m^2$-dimenstional)}
}

\end{frame}



\end{document}
