\documentclass[ignorenonframetext,11pt,color=option, mathserif]{beamer}
\usepackage{amsmath,amssymb}
%\usepackage{graphicx}
\usepackage{soul}
\usepackage{pgfarrows,pgfnodes}
\usepackage{url}
\usepackage{textcomp}
\usepackage[vcentermath]{youngtab}
\usepackage{multirow, multicol}
\usepackage{enumerate}
\usepackage{wrapfig}


%%%Beamer Specifications %%%------------------------------------------------------------
\usenavigationsymbolstemplate{}
\mode<presentation>
{
%  \usetheme{Malmoe}
  \useinnertheme{circles}
%  \setbeamertemplate{bibliography item}[text]

  \definecolor{basecolor}{rgb}{0.75, 0, 0}
  \colorlet{themecolor}{basecolor}
  \colorlet{alert}{basecolor}
  \usecolortheme[named= themecolor]{structure}


 
  \setbeamercolor{alerted text}{fg= alert}
  \setbeamercolor{example text}{fg= themecolor}
  \setbeamercolor{bibliography entry author}{fg=black!90!white}

  \setcounter{secnumdepth}{-1}
}
\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}


% COLOR
\usepackage{color}
\colorlet{dalert}{basecolor}
 \newcommand{\BLUE}[1]{{\color{blue}#1}}
 \newcommand{\GRAY}[1]{{\color{gray}#1}}
 \newcommand{\GR}[2]{{\only<#1>{\color{gray}} #2 }}
 
 
\definecolor{TypeB}{rgb}{0, 0, 0.85}
\definecolor{TypeA}{rgb}{0.5, 0, .75}
\definecolor{TypeA2}{rgb}{0.75, 0, 0}
\definecolor{Univ}{rgb}{0, 0.5, 0.2}

\newcommand{\GREEN}[1]{{\color{Univ}#1}}
%%%Theorem Environments %%%------------------------------------------------------------
\theoremstyle{plain}
\newtheorem{thm}{Theorem}%[section]
%\newtheorem{lemma}[thm]{Lemma}
%\newtheorem{fact}{Fact}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\theoremstyle{definition}
\newtheorem*{defn}{Definition}
\newtheorem{remark}[thm]{Remark}
\newtheorem*{remark*}{Remark}
\theoremstyle{example}
%\newtheorem*{example}{Example}
\newtheorem*{question}{Question}

\theoremstyle{remark}\newtheorem*{ack}{Acknowledgments}
\numberwithin{equation}{section}

%%% Characters %%%------------------------------------------------------------
%%%%% Alphabets %%%%% 
\def\cA{\mathcal{A}}\def\cB{\mathcal{B}}\def\cC{\mathcal{C}}\def\cD{\mathcal{D}}\def\cE{\mathcal{E}}\def\cF{\mathcal{F}}\def\cG{\mathcal{G}}\def\cH{\mathcal{H}}\def\cI{\mathcal{I}}\def\cJ{\mathcal{J}}\def\cK{\mathcal{K}}\def\cL{\mathcal{L}}\def\cM{\mathcal{M}}\def\cN{\mathcal{N}}\def\cO{\mathcal{O}}\def\cP{\mathcal{P}}\def\cQ{\mathcal{Q}}\def\cR{\mathcal{R}}\def\cS{\mathcal{S}}\def\cT{\mathcal{T}}\def\cU{\mathcal{U}}\def\cV{\mathcal{V}}\def\cW{\mathcal{W}}\def\cX{\mathcal{X}}\def\cY{\mathcal{Y}}\def\cZ{\mathcal{Z}}

\def\AA{\mathbb{A}} \def\BB{\mathbb{B}} \def\CC{\mathbb{C}} \def\DD{\mathbb{D}} \def\EE{\mathbb{E}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \def\II{\mathbb{I}} \def\JJ{\mathbb{J}} \def\KK{\mathbb{K}} \def\LL{\mathbb{L}} \def\MM{\mathbb{M}} \def\NN{\mathbb{N}} \def\OO{\mathbb{O}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\SS{\mathbb{S}} \def\TT{\mathbb{T}} \def\UU{\mathbb{U}} \def\VV{\mathbb{V}} \def\WW{\mathbb{W}} \def\XX{\mathbb{X}} \def\YY{\mathbb{Y}} \def\ZZ{\mathbb{Z}}  

\def\fa{\mathfrak{a}} \def\fb{\mathfrak{b}} \def\fc{\mathfrak{c}} \def\fd{\mathfrak{d}} \def\fe{\mathfrak{e}} \def\ff{\mathfrak{f}} \def\fg{\mathfrak{g}} \def\fh{\mathfrak{h}} \def\fj{\mathfrak{j}} \def\fk{\mathfrak{k}} \def\fl{\mathfrak{l}} \def\fm{\mathfrak{m}} \def\fn{\mathfrak{n}} \def\fo{\mathfrak{o}} \def\fp{\mathfrak{p}} \def\fq{\mathfrak{q}} \def\fr{\mathfrak{r}} \def\fs{\mathfrak{s}} \def\ft{\mathfrak{t}} \def\fu{\mathfrak{u}} \def\fv{\mathfrak{v}} \def\fw{\mathfrak{w}} \def\fx{\mathfrak{x}} \def\fy{\mathfrak{y}} \def\fz{\mathfrak{z}}
\def\fgl{\mathfrak{gl}}  \def\fsl{\mathfrak{sl}}  \def\fso{\mathfrak{so}}  \def\fsp{\mathfrak{sp}}  
\def\GL{\mathrm{GL}} \def\SL{\mathrm{SL}}  \def\Sp{\mathrm{Sp}}

\def\uu{\mathbf{u}}	\def\vv{\mathbf{v}}	\def\ww{\mathbf{w}}	\def\cc{\mathbf{c}}

\def\<{\langle} \def\>{\rangle}
\def\ad{\mathrm{ad}} 
\def\Aut{\mathrm{Aut}}
\def\dim{\mathrm{dim}} 
\def\End{\mathrm{End}} 
\def\ev{\mathrm{ev}} 
\def\half{\hbox{$\frac12$}}
\def\Hom{\mathrm{Hom}} 
\def\id{\mathrm{id}}
\def\img{\mathrm{img}}
\def\Ind{\mathrm{Ind}}
\def\ker{\mathrm{ker}}
\def\normeq{\unlhd}
\def\qtr{\mathrm{qtr}} 
\def\tr{\mathrm{tr}} 
\def\Tr{\mathrm{Tr}} 
\def\vep{\varepsilon}



\def\brac{]}

\def\one{\mathbbm{1}}

\newcommand{\BOX}[3]{
	\filldraw[white] (#1,#2)--(#1+1, #2)--(#1+1, #2+1)--(#1, #2+1)--(#1, #2);
	\draw[thick] (#1,#2)--(#1+1, #2)--(#1+1, #2+1)--(#1, #2+1)--(#1, #2);
	\node at (#1+.5,#2+.5){\tiny #3};
	}
\newcommand{\dBOX}[3]{
	\filldraw[black!20] (#1,#2)--(#1+1, #2)--(#1+1, #2+1)--(#1, #2+1)--(#1, #2);
	\draw (#1,#2)--(#1+1, #2)--(#1+1, #2+1)--(#1, #2+1)--(#1, #2);
	\node at (#1+.5,#2+.5){\tiny #3};
	}
\newcommand{\cBOX}[3]{
	\filldraw[#3] (#1,#2)--(#1+1, #2)--(#1+1, #2+1)--(#1, #2+1)--(#1, #2);
	\draw[thick] (#1,#2)--(#1+1, #2)--(#1+1, #2+1)--(#1, #2+1)--(#1, #2);
%	\node at (#1+.5,#2+.5){\tiny #3};
	}






%%%%%%%%%%%%%%%%
\usepackage{array}

\newsavebox\topalignbox
\newcolumntype{T}{%
  >{\begin{lrbox}\topalignbox
    \rule{0pt}{\ht\strutbox}}
  c
  <{\end{lrbox}%
    \raisebox{\dimexpr-\height+\ht\strutbox\relax}%
      {\usebox\topalignbox}}}
\newcolumntype{M}{>{$\vcenter\bgroup\hbox\bgroup}c<{\egroup\egroup$}}



%%%%%%%%%%%%%%%%%
\usepackage{tikz}
\def\SCALE{1}
\tikzstyle{V}=[draw, fill =black, circle, inner sep=0pt, minimum size=2pt]
\tikzstyle{sV}=[draw, fill =black, circle, inner sep=0pt, minimum size=1pt]
\tikzstyle{bV}=[draw, fill =black, circle, inner sep=0pt, minimum size=3pt]
\tikzstyle{wV}=[draw, fill =white, circle, inner sep=0pt, minimum size=3pt]
\tikzstyle{swV}=[draw, fill =white, circle, inner sep=0pt, minimum size=2pt]
\tikzstyle{over}=[draw=white,double=black,line width=2pt, double distance=.75pt]
\tikzstyle{diagram}=[line width=.75pt, scale=\SCALE]

\newcommand{\TikZ}[1]{\begin{matrix}\begin{tikzpicture}#1\end{tikzpicture}\end{matrix}}


\def\Over[#1,#2][#3,#4]{ %1,2=start position; 3,4=end position
	\draw[over]   (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);}
\def\Under[#1,#2][#3,#4]{ %1,2=start position; 3,4=end position
	\draw  (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);}
\def\Cross[#1,#2][#3,#4]{%Mimic over, under follows
	\Under[#3,#2][#1,#4]\Over[#1,#2][#3,#4]}


\def\Nodes[#1][#2]{
	 \foreach \x in {1,...,#2} {\filldraw [black] (\x,#1) circle (1.5pt);	}
	}

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

%% TikZ Commands particular to this paper:
\def\Pole[#1][#2,#3]{%1=horizontal location, 2 = bottom, 3=top
	\shade[left color=white,right color=white] (#1+.1,#2) rectangle (#1-.1,#3);
	\draw[over] (#1+.1,#2) to (#1+.1,#3) (#1-.1,#2) to (#1-.1,#3) ;}

\def\Tops[#1][#2][#3]{%1=pole locations, 2=top, 3=k 
	\foreach\x in {#1}{
		\draw (\x+.1,#2) -- (\x+.1,#2+.15) (\x-.1,#2) -- (\x-.1,#2+.15) ;
		\draw (\x+.1,#2+.15) arc (0:360:1mm and .5mm);}
	%Nodes 
	\foreach \x in {1,...,#3} {\draw (\x,#2)  to (\x,#2+.05) node[V]{};}
	}
\def\Bottoms[#1][#2][#3]{%1=pole locations, 2 = bottom, 3=top 
	\foreach\x in {#1}{
		\draw (\x+.1,#2) -- (\x+.1,#2-.1) (\x-.1,#2) -- (\x-.1,#2-.1) ;
		\draw (\x+.1,#2-.1) arc (0:-180:1mm and .5mm);}
	%Nodes 
	\foreach \x in {1,...,#3} {\draw (\x,#2)  to (\x,#2-.05) node[V]{};}
	}
\def\Caps[#1][#2,#3][#4]{%1=pole location, 2 = bottom, 3=top, 4=k 
	\Tops[#1][#3][#4]
	\Bottoms[#1][#2][#4]
	}

	
	
	
	
\def\PoleBottom[#1]{%1 = Vertical position
\draw (.35,#1) -- (.35,#1-.1) (.5,#1) -- (.5,#1-.1) ;
\draw (.5,#1-.1) arc (0:-180:2pt and 1.5pt);
}
\def\PoleTop[#1]{%1 = Vertical position
\draw (.35,#1) -- (.35,#1+.1) (.5,#1) -- (.5,#1+.1) ;
\draw (.5,#1+.1) arc (0:360:2pt and 1.5pt);
}
\def\PolesTop[#1]{%1 = Vertical position
\draw (.5,#1) -- (.5,#1+.1) (.35,#1) -- (.35,#1+.1) ;
\draw (.5,#1+.1) arc (0:360:2pt and 1.5pt);
\draw (-.15,#1) -- (-.15,#1+.1) (-.3,#1) -- (-.3,#1+.1) ;
\draw (-.15,#1+.1) arc (0:360:2pt and 1.5pt);
}
\def\PolesBottom[#1]{%1 = Vertical position
\draw (.5,#1) -- (.5,#1-.1) (.35,#1) -- (.35,#1-.1) ;
\draw (.5,#1-.1) arc (0:-180:2pt and 1.5pt);
\draw (-.15,#1) -- (-.15,#1-.1) (-.3,#1) -- (-.3,#1-.1) ;
\draw (-.15,#1-.1) arc (0:-180:2pt and 1.5pt);
}
\def\SymPolesTop[#1][#2]{%1 = Vertical position, 2=k
\draw (.5,#1) -- (.5,#1+.1) (.35,#1) -- (.35,#1+.1) ;
\draw (.5,#1+.1) arc (0:360:2pt and 1.5pt);
\draw (#2+.5,#1) -- (#2+.5,#1+.1) (#2+.65,#1) -- (#2+.65,#1+.1) ;
\draw (#2+.65,#1+.1) arc (0:360:2pt and 1.5pt);
}
\def\SymPolesBottom[#1][#2]{%1 = Vertical position, 2=k
\draw (.5,#1) -- (.5,#1-.1) (.35,#1) -- (.35,#1-.1) ;
\draw (.5,#1-.1) arc (0:-180:2pt and 1.5pt);
\draw (#2+.5,#1) -- (#2+.5,#1-.1) (#2+.65,#1) -- (#2+.65,#1-.1) ;
\draw (#2+.65,#1-.1)  arc (0:-180:2pt and 1.5pt);
}
\def\LPole[#1,#2]{%1 = bottom, 2=top
\shade[left color=white,right color=white] (-.3,#1) rectangle (-.15,#2);
\draw[over] (-.3,#1) -- (-.3,#2) (-.15,#1) -- (-.15,#2) ;
}
\def\RPole[#1,#2]{%1 = bottom, 2=top
\shade[left color=white,right color=white] (.35,#1) rectangle (.5,#2);
\draw[over] (.5,#1) -- (.5,#2) (.35,#1) -- (.35,#2) ;
}
\def\RRPole[#1,#2][#3]{%1 = bottom, 2=top
\shade[left color=white,right color=white] (#3+.65,#1) rectangle (#3+.5,#2);
\draw[over] (#3+.5,#1) -- (#3+.5,#2) (#3+.65,#1) -- (#3+.65,#2) ;
}
\def\PoleTwist[#1,#2]{%1 = bottom, 2=top
\draw (.15,#1) .. controls ++(0,#2*.5-#1*.5) and ++(0,-#2*.5+#1*.5) ..  (.5,#2) 
			(0,#1)   .. controls ++(0,#2*.55-#1*.55) and ++(0,-#2*.45+#1*.45) ..    (.35,#2) ;
\draw[thin,over] (.5,#1) .. controls ++(0,#2*.5-#1*.5) and ++(0,-#2*.5+#1*.5) ..  (.15,#2) 
		(.35,#1)  .. controls ++(0,#2*.45-#1*.45) and ++(0,-#2*.55+#1*.55) ..   (0,#2) ;
}
\def\Nodes[#1][#2]{
 \foreach \x in {1,...,#2} {
%       \draw (\x,0) circle (0.4cm);
	 \filldraw [black] (\x,#1) circle (2pt);
	 }
	}
%\def\Tops[#1][#2]{
% \foreach \x in {1,...,#2} {
%	%       \draw (\x,0) circle (0.4cm);
%	 \filldraw [black] (\x,#1+.4) circle (2.5pt);
%	 }
%	 \foreach \x in {1,...,#2} {
%	%       \draw (\x,0) circle (0.4cm);
%		 \draw (\x,#1) -- (\x,#1+.4) ;
%	 }
%}
%\def\Bottoms[#1][#2]{
% \foreach \x in {1,...,#2} {
%%       \draw (\x,0) circle (0.4cm);
%	 \filldraw [black] (\x,#1-.4) circle (2.5pt);
%	 }
%	 \foreach \x in {1,...,#2} {
%%       \draw (\x,0) circle (0.4cm);
%	 \draw (\x,#1) -- (\x,#1-.4) ;
%	 }
%	}
\def\Label[#1,#2][#3][#4]{%1,2 = top/bot position, 3=i, 4=label
	\node at (#3,#2+.4) {#4};
	\node at (#3,#1-.4) {#4};		}

\newcounter{r}
\newcommand\Part[1]{
        \setcounter{r}{1}
	 \foreach \x in {#1}{
 	{\ifnum\value{r}=1
		\draw (0,\value{r}-1)--(\x,\value{r}-1); 
		\fi}
	\draw (0,\value{r}) to (\x,\value{r});
   	\foreach \y in {0, ..., \x} {\draw (\y,\value{r})--(\y,\value{r}-1);}
	\addtocounter{r}{1}
 }}
 
  \def\PartUNIT{.125}
%Self-contained tikz images for \Part above. 
\newcommand{\PART}[1]{
\TikZ{[xscale=\PartUNIT, yscale=-\PartUNIT] 
	\Part{#1}
}
}


\def\UNIT{.7pt}
\def\SET{\setlength{\unitlength}{\UNIT}}
\def\LBIG{\color{white}\left|\color{black}}
\def\RBIG{\color{white}\right|\color{black}}
\def\XNG{\color{white}$\bullet$}


\def\REG[#1,#2]{
\draw[black,thick] (#1-.2,#2)--(#1+.2,#2) (#1-.15,#2-.15)--(#1+.15,#2+.15) (#1-.15,#2+.15)--(#1+.15,#2-.15) (#1,#2-.2)--(#1,#2+.2);
}
\def\REGe[#1,#2]{
\draw[black,thick] (#1,#2)--(#1+.2,#2) (#1,#2)--(#1+.15,#2+.15) (#1,#2)--(#1+.15,#2-.15) (#1,#2-.2)--(#1,#2+.2);
}
\def\REGnw[#1,#2]{
\draw[black,thick] (#1-.2,#2)--(#1,#2) (#1-.15,#2-.15)--(#1+.15,#2+.15) (#1-.15,#2+.15)--(#1,#2) (#1,#2)--(#1,#2+.2);
}
\def\sREG[#1,#2]{
\draw[black,thick] (#1-.13,#2)--(#1+.13,#2) (#1-.1,#2-.1)--(#1+.1,#2+.1) (#1-.1,#2+.1)--(#1+.1,#2-.1) (#1,#2-.13)--(#1,#2+.13);
}



\def\RowUNIT{.25}
\newcommand\Row[1]{
\begin{tikzpicture}[scale=\RowUNIT]
\draw(0,0) rectangle (4,1) (1,0) to (1,1) (2,0) to (2,1) (3,0) to (3,1);
	 \foreach \x [count = \c from 0] in {#1}{
		\node at (\c + .5, .5) {\tiny\x};
}
 \end{tikzpicture}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Brattelli diagram
\newcommand{\BoxPart}[2]{
\draw (0,0)--(5,0)--(5,4)--(0,4)--(0,0);
\coordinate (up) at (5,0); \coordinate (down) at (0,4);
\begin{scope}[shift=(up)]\Part{#1}\end{scope}
\begin{scope}[shift=(down)]\Part{#2}\end{scope}
}
\newcommand{\BoxPART}[2]{
\begin{tikzpicture}[xscale=\PartUNIT, yscale=-\PartUNIT]
	\node at (8,0) {}; \node at (0,7){};
	\BoxPart{#1}{#2}
\end{tikzpicture}
}
\newcommand{\BoxPARTs}[2]{
\begin{tikzpicture}[xscale=\PartUNIT, yscale=-\PartUNIT]
	\node at (8,0) {}; \node at (0,6){};
	\BoxPart{#1}{#2}
\end{tikzpicture}
}

\def\Frame[#1,#2][#3]{
	\draw[line width=1.5pt, dalert] (.5,#1) to (.5,#2) (#3*.75+.75,#1) to (#3*.75+.75,#2);
	\draw[densely dotted, dalert] (.5,#1) to (#3*.75+.75,#1) (.5,#2) to (#3*.75+.75,#2);
	\foreach \x in {1, ..., #3}{
		\node[V] (b\x) at (\x*.75+.25, #1){};	
		\node[V] (t\x) at (\x*.75+.25, #2) {};}}





\tikzstyle{EdgeLabel}=[blue!70!black, inner sep=2pt, fill=white]


\tikzstyle{FULL}=[draw, fill=red!10]
\tikzstyle{SumToL}=[draw, fill=red!10]
\tikzstyle{SumToBKL}=[draw, fill=red!10]
\tikzstyle{SumToA}=[draw, fill=red!10]
\tikzstyle{Overlapping}=[draw, fill=red!10]
\newcommand\TLNode[1]{ \node[#1] (\x\k) at (\x-.5*\k,\k){\scriptsize$\x$};}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\title{Two-boundary diagram algebras}
\author[Zajj Daugherty]{Zajj Daugherty \\
(Joint work in progress with Arun Ram)}
\date{October 17, 2017}





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



%%%%%%%%%%%%
%%%%%%%%%%%%
\begin{frame}{Motivating example: Schur-Weyl Duality}
The {\bf symmetric group} $S_k$ (permutations) as diagrams:

$$\begin{tikzpicture}[line width=.75pt]
%%%%%%%%%%%%%
\node at (.25,0){};%PLACE MARKER
\foreach \x in {1,...,5}{
\node[bV, label=above:{\scriptsize \color{themecolor} \x}](t\x) at (\x,1.5){}; 
\node[bV](b\x) at (\x,0){}; 
\uncover<1-2>{  \node[below] at (b\x) {\scriptsize \color{themecolor} \x};}}
\draw[->] (b2)--(t1); 
\draw[->] (b4)--(t2); 
\draw[->](b1)--(t3); 
\draw[->] (b5)--(t4); 
\draw[->] (b3)--(t5); 
%%%%%%%%%%%%%
\pgftransformyshift{-2.05cm}
%%%%%%%%%%%%%
\uncover<2->{
	\foreach \x in {1,...,5}{
		 \node[bV] (bt\x) at (\x,1.5){}; 
		 \node[bV, label=below:{\scriptsize \color{themecolor} \x}](bb\x) at (\x,0){}; 
	 }
	\draw[->] (bb2)--(bt1); 
	\draw[->] (bb1)--(bt2); 
	\draw[->] (bb4)--(bt3); 
	\draw[->] (bb5)--(bt4); 
	\draw[->] (bb3)--(bt5); 
}
%%%%%%%%%%%%%
\uncover<3->{
	\foreach \x in {1,...,5}{
	\draw[dotted, themecolor] (b\x)--(bt\x);
	}
}
%%%%%%%%%%%%%
\pgftransformyshift{1cm}
\pgftransformxshift{6cm}
%%%%%%%%%%%%%
\uncover<4->{\node at (0,.75) {$=$};
	\foreach \x in {1,...,5}{
		 \node[bV, label=above:{\scriptsize \color{themecolor} \x}] (rt\x) at (\x,1.5){}; 
		 \node[bV, label=below:{\scriptsize \color{themecolor} \x}] (rb\x) at (\x,0){}; 
	 }
	\draw[->] (rb1)--(rt1); 
	\draw[->] (rb5)--(rt2); 
	\draw[->] (rb2)--(rt3); 
	\draw[->] (rb3)--(rt4); 
	\draw[->] (rb4)--(rt5); 
}
\end{tikzpicture}$$
\uncover<2->{(with multiplication given by concatenation)}
\end{frame}

%%%%%%%%%%%%
%%%%%%%%%%%%


\begin{frame}{Motivating example: Schur-Weyl Duality}
\begin{enumerate}[] 
\item  $\GL_n(\CC)$ acts on $\CC^n \otimes \CC^n \otimes \cdots \otimes \CC^n = \left( \CC^n\right)^{\otimes k}$
diagonally.
$$\BLUE{g} \cdot (v_1 \otimes v_2 \otimes \cdots \otimes v_k) = \BLUE{g} v_1 \otimes \BLUE{g} v_2 \otimes \cdots \otimes \BLUE{g} v_k.$$
\item<2-> $S_k$ also acts on  $\left( \CC^n\right)^{\otimes k}$ by place permutation.
$$\begin{tikzpicture}[line width=.75pt, scale=.9]
\foreach \x in {1,...,5}{
\node[bV](t\x) at (\x,1){}; 
\node[bV, label=below:{\small \color{themecolor} $v_{\x}$}](b\x) at (\x,0){}; 
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \color{themecolor} $\otimes$};
\node at (\x+.5, -.3) {\small \color{themecolor} $\otimes$};
}
\node[above] at (t1) {\small \color{themecolor}$v_{2}$};
\node[above] at (t2) {\small \color{themecolor} $v_{4}$};
\node[above] at (t3) {\small \color{themecolor} $v_{1}$};
\node[above] at (t4) {\small \color{themecolor} $v_{5}$};
\node[above] at (t5) {\small \color{themecolor} $v_{3}$};
\draw[->] (b2)--(t1); 
\draw[->] (b4)--(t2);
\draw[->] (b1)--(t3); 
\draw[->] (b5)--(t4); 
\draw[->] (b3)--(t5); 
\end{tikzpicture}$$
\item<3-> These actions commute!  
$$\begin{minipage}{4.5cm}\begin{tikzpicture}[line width=.75pt, scale=.9]
\foreach \x in {1,...,5}{
\node[bV](t\x) at (\x,1){}; 
\node[bV, label=below:{\small \color{blue}$g$\alert{$v_{\x}$}}](b\x) at (\x,0){}; 
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \color{themecolor} $\otimes$};
\node at (\x+.5, -.3) {\small \color{themecolor} $\otimes$};
}
\node[above] at (t1) {\small \color{blue}$g$\alert{$v_{2}$}};
\node[above] at (t2) {\small \color{blue}$g$\alert{$v_{4}$}};
\node[above] at (t3) {\small \color{blue}$g$\alert{$v_{1}$}};
\node[above] at (t4) {\small \color{blue}$g$\alert{$v_{5}$}};
\node[above] at (t5) {\small \color{blue}$g$\alert{$v_{3}$}};
\draw[->] (b2)--(t1); 
\draw[->] (b4)--(t2);
\draw[->] (b1)--(t3); 
\draw[->] (b5)--(t4); 
\draw[->] (b3)--(t5); 
\end{tikzpicture}\end{minipage} \text{ vs. } 
\begin{minipage}{4.5cm}\begin{tikzpicture}[line width=.75pt, scale=.9]
\foreach \x in {1,...,5}{
\node[bV](t\x) at (\x,1){}; 
\node[bV, label=below:{\small \alert{$v_{\x}$}}](b\x) at (\x,0){}; 
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \alert{$\otimes$}};
\node at (\x+.5, -.3) {\small \alert{$\otimes$}};
}
\node[above] at (t1) {\small \color{blue}$g$\alert{$v_{2}$}};
\node[above] at (t2) {\small \color{blue}$g$\alert{$v_{4}$}};
\node[above] at (t3) {\small \color{blue}$g$\alert{$v_{1}$}};
\node[above] at (t4) {\small \color{blue}$g$\alert{$v_{5}$}};
\node[above] at (t5) {\small \color{blue}$g$\alert{$v_{3}$}};
\draw[->] (b2)--(t1); 
\draw[->] (b4)--(t2);
\draw[->] (b1)--(t3); 
\draw[->] (b5)--(t4); 
\draw[->] (b3)--(t5); \end{tikzpicture}\end{minipage}$$

\end{enumerate}
\end{frame}

%%%%%%%%%%%%
%%%%%%%%%%%%



\begin{frame}{{Motivating example: Schur-Weyl Duality}}
Schur (1901):
$$\!\!\!\underbrace{\End_{\GL_n}\left(\left( \CC^n\right)^{\otimes k} \right)}_{\text{\scriptsize (all linear maps that} \atop \text{\scriptsize commute with $\GL_n$)}} 
	= \underbrace{\pi(\CC S_k)\phantom{\Big|}\!\!}_{\text{\scriptsize (img of $S_k$} \atop\text{\scriptsize action)}}
	 \quad \text{ and } \quad \End_{S_k}\left( \left( \CC^n\right)^{\otimes k}\right) 
	 = \underbrace{\rho(\CC \GL_n)\phantom{\Big|}\!\!}_{\text{\scriptsize (img of $\GL_n$} \atop\text{\scriptsize action)}}
\!\!. $$%
\uncover<2->{\begin{block}{Powerful consequence: a duality between representations}
		The double-centralizer relationship produces
			$$\left( \CC^n\right)^{\otimes k}  \cong \bigoplus_{\tiny \lambda \vdash k }  G^\lambda \otimes S^\lambda
			\quad \text{ as a $\GL_n$-$S_k$ bimodule,}$$
where 		\begin{tabular}{rcl}
			$G^\lambda$ &are distinct irreducible& $\GL_n$-modules\\
			$S^\lambda$ &are distinct irreducible& $S_k$-modules\\
			\end{tabular}
	\end{block}}
\end{frame}


%%%%%%%%%%%%
%%%%%%%%%%%%



\begin{frame}{More centralizer algebras}

\hbox{
\hspace{-.1in}
\begin{minipage}[t]{2in}
Brauer (1937) \\
\small Orthogonal and symplectic groups \only<1->{(and Lie algebras)} acting on $(\CC^n)^{\otimes k}$ diagonally
centralize\\ the \textbf{Brauer algebra}:\\
\begin{tikzpicture}[line width=.75pt, scale=.75]
	\foreach\x in {1,...,5}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left] (t2) to (t1) (b2) to (b3);
	\draw (b1)--(t3) (b5)--(t4) (b4)--(t5);
	\foreach\x/\y in {1/a,2/b,3/c,4/d,5/e}{\node[below] at (b\x) {\small\alert{$v_{\y}$}};}
	\node[above] at (t1) {\small\alert{$v_i$}};
	\node[above] at (t2) {\small\alert{$v_i$}};
	\node[above] at (t3) {\small\alert{$v_a$}};
	\node[above] at (t4) {\small\alert{$v_d$}};
	\node[above] at (t5) {\small\alert{$v_d$}};
	\foreach\x in {1,...,4}{
		\node[above] at (\x+.5,1) {\small\alert{$\otimes$}};
		\node[below] at (\x+.5,0) {\small\alert{$\otimes$}};}
	\node[left] at (1,1.4) {\small\alert{$\displaystyle \delta_{b,c} \sum_{i=1}^n$}};
	\draw[|->,themecolor] (5.5,0)--(5.5,1);
	\foreach\x in {1,...,5}{\node[bV] at (t\x){};\node[bV] at (b\x){};}
\end{tikzpicture}

\centerline{with 
$\TikZ{[line width=.75pt, scale=.75]
	\draw (0,0) ellipse (15pt and 7pt);
}
	= n$}
\end{minipage}\qquad
\uncover<2->{\begin{minipage}[t]{2.1in}
Temperley-Lieb (1971) \\
\small $\GL_2$ and $\SL_2$ \only<1->{(and $\fgl_2$ and $\fsl_2$)} acting on $(\CC^2)^{\otimes k}$
diagonally centralize\only<1->{\\}
 the \textbf{Temperley-Lieb algebra}:\\~\\
\begin{tikzpicture}[line width=.75pt, scale=.75]
	\foreach\x in {1,...,5}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left] (t3) to (t2) (b3) to (b4);
	\draw (b1)--(t1) (b5)--(t5) (b2)--(t4);
	\foreach\x/\y in {1/a,2/b,3/c,4/d,5/e}{\node[below] at (b\x) {\small\alert{$v_{\y}$}};}
	\node[above] at (t1) {\small\alert{$v_a$}};
	\node[above] at (t2) {\small\alert{$v_i$}};
	\node[above] at (t3) {\small\alert{$v_i$}};
	\node[above] at (t4) {\small\alert{$v_b$}};
	\node[above] at (t5) {\small\alert{$v_e$}};
	\foreach\x in {1,...,4}{
		\node[above] at (\x+.5,1) {\small\alert{$\otimes$}};
		\node[below] at (\x+.5,0) {\small\alert{$\otimes$}};}
	\node[left] at (1,1.4) {\small\alert{$\displaystyle \delta_{c,d} \sum_{i=1}^2$}};
	\draw[|->,themecolor] (5.5,0)--(5.5,1);
	\foreach\x in {1,...,5}{\node[bV] at (t\x){};\node[bV] at (b\x){};}
\end{tikzpicture}

\centerline{with 
$\TikZ{[line width=.75pt, scale=.75]
	\draw (0,0) ellipse (15pt and 7pt);
}
	= 2$}}
\end{minipage}}

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

(Diagrams encode maps $V^{\otimes k} \to V^{\otimes k}$ that commute with the action of some classical algebra.) 

\end{frame}


%%%%%%%%%%%
%%%%%%%%%%%


\begin{frame}{Quantum groups and braids}
\def\TOP{1.5}
\small
Fix $q \in \CC$, and let  $\cU=\cU_q\fg$ be the  Drinfeld-Jimbo quantum group associated to Lie algebra $\fg$. \pause

$\cU\otimes \cU$ has an invertible element $\cR=\sum_{\cR} R_1\otimes R_2$ that yields a map\\
\centerline{\begin{tikzpicture}[scale=.8, line width=.75pt ]
	\node[left] at (0,.375) 
		{$\check \cR_{VW}\colon V\otimes W \longrightarrow W\otimes V$};
	\Under[1,0][2,.75] 	\Over[1,.75][2,0]
	\Nodes[.75][2]	\Nodes[0][2]
	\node at (1.5,1) {$W \otimes V$};
	\node at (1.5,-.5) {$V \otimes W$};
\end{tikzpicture}}
that \quad 
	\alert{(1)} satisfies braid relations,  and \\ 
\hspace{.375in}	\alert{(2)} commutes with the  $\cU$-action on $V\otimes W$\\
for any $\cU$-module $V$. \pause

~\\

The \only<4>{\alert{one-pole/affine} }\only<5>{\alert{two-pole} }braid group shares
 a commuting action\\ with $\cU$ on $\only<4->{\alert{M \otimes}}V^{\otimes k}\only<5->{\alert{\otimes N}}$:
$$\TikZ{[line width=.75pt]
	\draw (2,1.5).. controls (4,1) .. (5,0);
	\Under[1,1.5][1.75,.75];
	\uncover<3-4>{\Under[4,1.5][4.5,.75];}
	\uncover<5>{\Under[4,1.5][5.8,.75];}
	\uncover<3>{
		\Under[1.25,.75][2,0];
		\Over[3,1.5][1.25,.75];}
	\uncover<4->{
		\Under[.2,.75][2,0];
		\Pole[.45][0,\TOP]
		\Over[3,1.5][.2,.75];}
	\uncover<5>{\Pole[5.55][0,\TOP]}
	\Over[5,1.5][4,.75];
	\Over[1.75,.75][1,0];
	\Over[4,.75][4,0];
	\uncover<3-4>{\Over[4.5,.75][3,0];}
	\uncover<5>{\Over[5.8,.75][3,0];}
\foreach \x in {1,...,5}{
	\draw (\x,0)  to (\x,-.05) node[bV, label=below:{\alert{$V$}}]{};
	\draw (\x,\TOP)  to (\x,\TOP+.05) node[bV, label=above:{\alert{$V$}}]{};
	\ifnum \x<5 
		\node at (\x+.5,1.86) {\alert{$\otimes$}};
		\node at (\x+.5,-.4) {\alert{$\otimes$}}; \fi
	}
\uncover<4->{
	\node at (.55,1.86) {\alert{$M \otimes$}};
	\node at (.55,-.4) {\alert{$M \otimes$}};}
\uncover<5>{
	\node at (5.45,1.86) {\alert{$\otimes N$}};
	\node at (5.45,-.4) {\alert{$\otimes N$}};}
\uncover<4>{
	\Caps[.45][0,\TOP][5]}
\uncover<5>{
	\Caps[.45,5.55][0,\TOP][5]}
}
\qquad \quad
\uncover<4->{\begin{matrix}\text{\scriptsize Around the pole:}\\
\TikZ{[line width=.75pt]
	\Under[1,.9][.2,.5]
	\Pole[.5][0.2,0.8] 
	\Over[1,.1][.2,.5]
	\node[above] at (.75,.9) {\small$M \!\!\otimes\! V$}; 
	\node[below] at (.75,.1) {\small$M \!\!\otimes\! V$}; 
	\Caps[.5][0.2,0.8][1]
	}
=  \check R_{MV} \check R_{VM} \end{matrix}}
$$

\end{frame}

%%%%%%%%%%%%
%%%%%%%%%%%%



%\begin{frame}%TABLE OF DIAGRAM ALGEBRAS Part A
%\hbox{\hspace{-.4in}\includegraphics{diagrams-slideA}}
%\end{frame}
%
%\begin{frame}%TABLE OF DIAGRAM ALGEBRAS Part B
%\hbox{\hspace{-.4in}\includegraphics{diagrams-slideB}}
%\end{frame}

\begin{frame}%TABLE OF DIAGRAM ALGEBRAS FULL
\hbox{\hspace{-.4in}\includegraphics{diagrams-slideC}}
\end{frame}

\begin{frame}%TABLE OF DIAGRAM ALGEBRAS TL
\hbox{\hspace{-.4in}\includegraphics{diagrams-slideTL}}
\end{frame}









\begin{frame}{Two-boundary Temperley-Lieb algebras}
\def\SCALE{.5}
%{\bf Nienhuis, De Gier, Batchelor (2004):} \\
[MNGB04] Fix $z, \delta_0, \delta_k \in \CC$. The \emph{two-boundary Temperley-Lieb algebra $TL_k$} is a diagram algebra generated over $\CC$ by diagrams 
$$e_0 = 
\TikZ{[diagram]
	\Frame[0,1.5][5]
	\node[above] at (t1) {\tiny$1$}; \node[below] at (b1) {\tiny$1$}; 
	\foreach \x in {2,3,4,5}{\draw (b\x) to (t\x);}
	\node[wV] (L1) at (.5,.5) {}; 
	\node[wV] (L2) at (.5,1) {}; 
	\draw[bend right=50] (L2) to (t1) (b1) to (L1); 
	}, \quad 
e_k = 
\TikZ{[diagram]
	\Frame[0,1.5][5]
	\node[above] at (t5) {\tiny$k$}; \node[below] at (b5) {\tiny$k$}; 
	\foreach \x in {1,2,3,4}{\draw (b\x) to (t\x);}
	\node[wV] (R1) at (4.5,.5) {}; 
	\node[wV] (R2) at (4.5,1) {}; 
	\draw[bend left=50] (R2) to (t5) (b5) to (R1); 
	},
\quad \text{ and } \quad 
e_i = 
\TikZ{[diagram]
	\Frame[0,1.5][5]
	\node[above] at (t2) {\tiny$i$}; \node[below] at (b2) {\tiny$i$}; 
	\foreach \x in {1,4,5}{\draw (b\x) to (t\x);}
	\draw[bend right=70] (t2) to (t3) (b3) to (b2); 
	}
	$$
for  $i=1, \dots, k-1$\uncover<2->{,
	with relations $e_i e_j = e_j e_i$ for $|i-j|>1$, }
\def\SCALE{.3}
$$\hspace{-.1in}\begin{array}{c@{\qquad}c}
\uncover<2->{\begin{matrix}e_i e_{i \pm 1} e_i = e_i \\ \text{\scriptsize for  $1 \leq i \leq k-1$,} \end{matrix}}
&
\uncover<3->{
\fbox{ $
	\TikZ{[diagram]
	\draw (3,0) to (3,1) (1,1) to (1,2) (3,2) to (3,3);
	\draw[bend left=50]  (1,0) to (2,0) (2,1) to (1,1) (2,1) to (3,1) (3,2) to (2,2) (1,2) to (2,2) (2,3) to (1,3);
	}
	=
	\TikZ{[diagram]
	\draw (3,0) to (3,1.5);
	\draw[bend left=80]  (1,0) to (2,0) (2,1.5) to (1,1.5);
	}$\ }
~\text{ or }~ 
\fbox{ $
	\TikZ{[diagram]
	\draw (2,1) to (2,2);
	\draw[bend left=50]  (1,0) to (2,0) (2,1) to (1,1) (1,2) to (2,2) (2,3) to (1,3);
	\draw[line width=1.5pt, dalert] (.5,-.1)--(.5,3.1);
		\node[wV] (L1) at (.5,1.25) {}; 
		\node[wV] (L2) at (.5,1.75) {}; 
	\draw [bend left=40] (1,2) to (L2) (L1) to (1,1);
	}
	=
	\TikZ{[diagram]
	\draw[line width=1.5pt, dalert] (.5,-.1)--(.5,1.6);
	\draw[bend left=80]  (1,0) to (2,0) (2,1.5) to (1,1.5);
	}$\ }
~\text{ or }~
\fbox{ $
	\TikZ{[diagram]
	\draw (1,1) to (1,2);
	\draw[bend left=50]  (1,0) to (2,0) (2,1) to (1,1) (1,2) to (2,2) (2,3) to (1,3);
	\draw[line width=1.5pt, dalert] (2.5,-.1)--(2.5,3.1);
		\node[wV] (R1) at (2.5,1.25) {}; 
		\node[wV] (R2) at (2.5,1.75) {}; 
	\draw [bend right=40] (2,2) to (R2) (R1) to (2,1);
	}
	=
	\TikZ{[diagram]
	\draw[line width=1.5pt, dalert] (2.5,-.1)--(2.5,1.6);
	\draw[bend left=80]  (1,0) to (2,0) (2,1.5) to (1,1.5);
	}$\ }}
\\~\\
\uncover<2->{e_i^2 = \delta_i e_i. }
& 
\uncover<4->{\fbox{$\TikZ{[diagram]
\draw[bend left=80]  (1,0) to (2,0) (2,1) to (1,1) (1,1) to (2,1) (2,2) to (1,2);
}
=
\delta~
\TikZ{[diagram]
\draw[bend left=80]  (1,0) to (2,0) (2,1.5) to (1,1.5);
}$}}
\uncover<4->{	
	~\text{ or }~ 
	\fbox{$
		\TikZ{[diagram]
		\draw[line width=1.5pt, dalert] (.25,-.1)--(.25,3.1);
			\node[wV] (L1) at (.25,2.5) {}; 
			\node[wV] (L2) at (.25,2) {}; 
			\node[wV] (L3) at (.25,1) {}; 
			\node[wV] (L4) at (.25,.5) {}; 
		\draw [bend left=50] (1,3) to (L1) (L2) to (1,1.5) to (L3) (L4) to (1,0);
		}
		= \delta_0~
		\TikZ{[diagram]
		\draw[line width=1.5pt, dalert] (.25,-.1)--(.25,1.6);
			\node[wV] (L3) at (.25,1) {}; 
			\node[wV] (L4) at (.25,.5) {}; 
		\draw [bend left=50] (1,1.5) to (L3) (L4) to (1,0);
		}$}
	~\text{ or }~ 
	\fbox{$
		\TikZ{[diagram]
		\draw[line width=1.5pt, dalert] (1.75,-.1)--(1.75,3.1);
			\node[wV] (R1) at (1.75,2.5) {}; 
			\node[wV] (R2) at (1.75,2) {}; 
			\node[wV] (R3) at (1.75,1) {}; 
			\node[wV] (R4) at (1.75,.5) {}; 
		\draw [bend right=50] (1,3) to (R1) (R2) to (1,1.5) to (R3) (R4) to (1,0);
		}
		= \delta_k~
		\TikZ{[diagram]
		\draw[line width=1.5pt, dalert] (1.75,-.1)--(1.75,1.6);
			\node[wV] (R3) at (1.75,1) {}; 
			\node[wV] (R4) at (1.75,.5) {}; 
		\draw [bend right=50] (1,1.5) to (R3) (R4) to (1,0);
		}$}}
\end{array}$$
\uncover<5->{\small (Side loops are resolved with a $1$ or a $\delta_i$ depending on whether there are an even or odd  number of connections below their lowest point.)}
\end{frame}




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

\def\SCALE{.75}
\alert{Diagram multiplication:}
$$\begin{tikzpicture}[diagram, scale=.75]
%%%%%%%%%%%%%
\node at (.25,0){};%PLACE MARKER
\draw[line width=1.5pt, dalert] (.5,0) to (.5,1.5) (5.5,0) to (5.5,1.5);
\draw[densely dotted, dalert] (.5,0) to (5.5,0) (.5,1.5) to (5.5,1.5);
\foreach \x in {1,...,5}{
\node[V](t\x) at (\x,1.5){}; 
\node[V](b\x) at (\x,0){}; }
\node[wV] (R1) at (5.5,1) {}; 
\node[wV] (R2) at (5.5,.5) {}; 
\draw (t1) to [bend right] (t2)(t4) to [bend right] (t5) (b1) to (t3);
\only<1-3>{\draw (b2) to [bend left] (b3) (b4) to [bend left] (R1) (b5) to [bend left] (R2);}
\only<4->{\draw[red] (b4) to [bend left] (R1);
		\draw[blue] (b5) to [bend left] (R2);
		\draw[red!50!yellow] (b2) to [bend left] (b3);}
%%%%%%%%%%%%
\pgftransformyshift{-2.05cm}
%%%%%%%%%%%%%
\uncover<2->{
\draw[line width=1.5pt, dalert] (.5,0) to (.5,1.5) (5.5,0) to (5.5,1.5);
\draw[densely dotted, dalert] (.5,0) to (5.5,0) (.5,1.5) to (5.5,1.5);
	\foreach \x in {1,...,5}{
		 \node[V] (bt\x) at (\x,1.5){}; 
		 \node[V](bb\x) at (\x,0){}; 
	 }
\node[wV] (R1) at (5.5,1) {}; 
\node[wV] (R2) at (5.5,.5) {}; 
\draw   (bt1) to (bb5) (bb1) to [bend left] (bb2) (bb3) to [bend left] (bb4);
\only<1-3>{\draw (bt2) to [bend right] (bt3) (bt4) to [bend right] (R2) (bt5) to [bend right] (R1);}
\only<4->{\draw[red] (bt4) to [bend right] (R2);
		\draw[blue] (bt5) to [bend right] (R1);
		\draw[red!50!yellow] (bt2) to [bend right] (bt3);}
}
%%%%%%%%%%%%%
\only<3>{\foreach \x in {1,...,5}{\draw[structure] (b\x)--(bt\x);}}
\only<4->{\draw[structure] (b1)--(bt1);
		\draw[red] (b4)--(bt4);
		\draw[blue] (b5)--(bt5);
		\draw[red!50!yellow] (b2)--(bt2) (b3)--(bt3);}
%%%%%%%%%%%%%
\pgftransformyshift{1cm}
\pgftransformxshift{6cm}
%%%%%%%%%%%%%
\uncover<5->{\node at (0,.75) {$=$};
\draw[line width=1.5pt, dalert] (.5,0) to (.5,1.5) (5.5,0) to (5.5,1.5);
\draw[densely dotted, dalert] (.5,0) to (5.5,0) (.5,1.5) to (5.5,1.5);
	\foreach \y [count=\x from 1] in {1,5,2,3,4}{
		 \node[V] (rt\x) at (\x,1.5){}; 
		 \node[V] (rb\x) at (\x,0){}; 
	 }
	\draw(rt1) to [bend right] (rt2) (rt4) to [bend right] (rt5)  (rt3)--(rb5)
		(rb1) to [bend left] (rb2) (rb3) to [bend left] (rb4);
	\node[right] at (5.5,.75) {$*{\color{red!50!yellow}\delta}*{\color{red}1}*{\color{blue}\delta_k} $};
}
\end{tikzpicture}$$\pause\pause\pause\pause\pause
\noindent In short, $TL_k$ has basis given by non-crossing diagrams with 
\begin{enumerate}[(1)]
	\item $k$ connections to the top and to the bottom,
	\item an even number of connections to the right and to the left, and 
	\item no edges with both ends on the left or both ends on the right. 
\end{enumerate}\pause However, 
\def\SCALE{.6}
$$
\TikZ{[diagram]
\draw[line width=1.5pt, dalert] (.5,-.3)--(.5,1.85)  (4.5,-.3)--(4.5,1.85);
\draw[densely dotted, dalert] (.5,-.3)--(4.5,-.3) (.5,1.85) --(4.5,1.85);
\foreach \x in {1,...,5}{\node[V] (b\x) at (\x*.75+.25, -.25) {}; \node[V] (t\x) at (\x*.75+.25, 1.75){};}
\foreach \y in {1,2,4,5}{\node[wV] (l\y) at (.5, \y*.25){}; \node[wV] (r\y) at (4.5, \y*.25){}; \draw (l\y)--(r\y);}
\node[wV] (r6) at (4.5, 1.55){}; \node[wV] (r0) at (4.5, -.05){}; 
\draw[bend right=60] 
		(t1) to  (t2)	(t3) to  (t4)	(b4) to  (b3)	(b2) to  (b1) ;
\draw[bend right=40] 	(t5) to (r6) (r0) to (b5);
\foreach \x in {-1,0,1}{
		\node[draw, fill =black, circle, inner sep=0pt, minimum size=.25pt] at (2.5, .75+.125*\x) {};}
\draw[|-|] (0,.1) to node[left, midway] {$2\ell$} (0,1.4);
} \in TL_k$$
So unlike the classical T-L algebras, $TL_k$ is not finite dimensional! 
\pause Take quotient giving \\ 
\centerline{$\displaystyle
\TikZ{[diagram]
\draw[line width=1.5pt] (.5,0)--(.5,.75);
\draw[line width=1.5pt] (2.5,0)--(2.5,.75);
\node[wV] (l1) at (.5, .2){}; \node[wV] (r1) at (2.5, .2){}; \draw (l1)--(r1);
\node[wV] (l2) at (.5, .55){}; \node[wV] (r2) at (2.5, .55){}; \draw (l2)--(r2);
\only<1-4>{\foreach\x in {.5,2.5}{\foreach\y in {-.15,.9}{\foreach \d in {-1,0,1}{
		\node[draw, fill =black, circle, inner sep=0pt, minimum size=.25pt] at (\x, \y+.1*\d) {};}}}}
} = z
$}
\end{frame}



\begin{frame}{Representation theory of $TL_k$: action on diagrams}
\def\BOTTOM{
		\only<1-8>{\draw (1,-2) node[sV]{} to [bend right] (.5,-1.5) node[swV]{}
			(2,-2) node[sV]{} to [bend right] (.5,-1.1) node[swV]{}
			(3,-2) node[sV]{} to [bend right] (.5,-.7) node[swV]{}
			(4,-2) node[sV]{} to [bend right] (.5,-.3) node[swV]{};}
		\only<9>{\draw[red] (1,-2) node[sV]{} to [bend right] (.5,-1.5) node[swV]{}
			(2,-2) node[sV]{} to [bend right] (.5,-1.1) node[swV]{}
			(3,-2) node[sV]{} to [bend right] (.5,-.7) node[swV]{}
			(4,-2) node[sV]{} to [bend right] (.5,-.3) node[swV]{};}}

$$d= \TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,2) to (.5, -2) (4.5,2) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
			\BOTTOM
		}
$$

\pause

$$\hspace{-.5cm}e_1 d= \TikZ{[scale = .45]
		\draw[dotted] (.5,3) to  (4.5,3) (.5,2) to (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,3) to (.5, -2) (4.5,3) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		\BOTTOM
		\begin{scope}[every node/.style={sV}]
		\draw[bend left=80] (1,2) to  (2,2)  (2,3) node{} to (1,3)node{};
		\draw (3,2) to (3,3) node{} (4,2) to (4,3) node{};
		\end{scope}
		}\pause
		= \delta_0
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,2) to (.5, -2) (4.5,2) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend right=70] (2,2) node[sV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		\BOTTOM
		}
\qquad \pause
e_4 d= \TikZ{[scale = .45]
		\draw[dotted] (.5,3) to  (4.5,3) (.5,2) to (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,3) to (.5, -2) (4.5,3) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		\BOTTOM
		\begin{scope}[every node/.style={sV}]
		\draw[bend right] (4,3) node{} to  (4.5,2.7) node[swV]{}  (4.5, 2.3) node[swV]{} to (4,2) node{};
		\draw (3,2) to (3,3) node{} (2,2) to (2,3) node{} (1,2) to (1,3) node{};
		\end{scope}
		}\pause
		= 
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,2) to (.5, -2) (4.5,2) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend right] (4.5,1.5) node[swV]{}
			(.5,.3) node[swV]{} to (4.5,.3) node[swV]{};
		\BOTTOM
		}
$$\pause
$$e_3e_4 d= \TikZ{[scale = .45]
		\draw[dotted] (.5,4) to  (4.5,4)  (.5,3) to  (4.5,3) (.5,2) to (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,4) to (.5, -2) (4.5,4) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		\BOTTOM
		\begin{scope}[every node/.style={sV}]
		\draw[bend right] (4,3) node{} to  (4.5,2.7) node[swV]{}  (4.5, 2.3) node[swV]{} to (4,2) node{};
		\draw (3,2) to (3,3) node{} (2,2) to (2,3) node{} (1,2) to (1,3) node{};
		\draw[bend left=80] (3,3) to  (4,3)  (4,4) node{} to (3,4)node{};
		\draw (1,3) to (1,4) node{} (2,3) to (2,4) node{};
		\end{scope}
		}\pause
		= 
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,2) to (.5, -2) (4.5,2) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend right=80] (4,2) node[sV]{}
			(.5,.7) node[swV]{} to  (4.5,.7) node[swV]{}
			(.5,.3) node[swV]{} to  (4.5,.3) node[swV]{}
			;
		\BOTTOM
		}\pause
		=
		z
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,-2) to (4.5,-2);
		\draw[line width=1pt] (.5,2) to (.5, -2) (4.5,2) to (4.5,-2);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend right=80] (4,2) node[sV]{}
			;
		\BOTTOM
		}
		$$
		
%(using \centerline{$\displaystyle
%\TikZ{[diagram]
%\draw[line width=1.5pt] (.5,0)--(.5,.75);
%\draw[line width=1.5pt] (2.5,0)--(2.5,.75);
%\node[wV] (l1) at (.5, .2){}; \node[wV] (r1) at (2.5, .2){}; \draw (l1)--(r1);
%\node[wV] (l2) at (.5, .55){}; \node[wV] (r2) at (2.5, .55){}; \draw (l2)--(r2);
%\only<1-4>{\foreach\x in {.5,2.5}{\foreach\y in {-.15,.9}{\foreach \d in {-1,0,1}{
%		\node[draw, fill =black, circle, inner sep=0pt, minimum size=.25pt] at (\x, \y+.1*\d) {};}}}}
%} = z
%$)}
\end{frame}

\begin{frame}{Representation theory of $TL_k$: half diagrams}

$$d= \TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,0) to (4.5,0);
		\draw[line width=1pt] (.5,2) to (.5, 0) (4.5,2) to (4.5,0);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		}
$$



$$\hspace{-.5cm}e_1 d= \TikZ{[scale = .45]
		\draw[dotted] (.5,3) to  (4.5,3) (.5,2) to (4.5,2) (.5, 0) to (4.5, 0);
		\draw[line width=1pt] (.5,3) to (.5, 0) (4.5,3) to (4.5, 0);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		\begin{scope}[every node/.style={sV}]
		\draw[bend left=80] (1,2) to  (2,2)  (2,3) node{} to (1,3)node{};
		\draw (3,2) to (3,3) node{} (4,2) to (4,3) node{};
		\end{scope}
		}
		= \delta_0
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,0) to (4.5,0);
		\draw[line width=1pt] (.5,2) to (.5, 0) (4.5,2) to (4.5,0);
		\draw (1,2) node[sV]{} to [bend right=70] (2,2) node[sV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		}
\qquad 
e_4 d= \TikZ{[scale = .45]
		\draw[dotted] (.5,3) to  (4.5,3) (.5,2) to (4.5,2) (.5, 0) to (4.5, 0);
		\draw[line width=1pt] (.5,3) to (.5, 0) (4.5,3) to (4.5, 0);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		\begin{scope}[every node/.style={sV}]
		\draw[bend right] (4,3) node{} to  (4.5,2.7) node[swV]{}  (4.5, 2.3) node[swV]{} to (4,2) node{};
		\draw (3,2) to (3,3) node{} (2,2) to (2,3) node{} (1,2) to (1,3) node{};
		\end{scope}
		}
		= 
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,0) to (4.5,0);
		\draw[line width=1pt] (.5,2) to (.5, 0) (4.5,2) to (4.5,0);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend right] (4.5,1.5) node[swV]{};
		\only<1>{\draw (.5,.3) node[swV]{} to (4.5,.3) node[swV]{};}
		\only<2>{\draw[red] (.5,.3) node[swV]{} to (4.5,.3) node[swV]{};}
		\only<3>{}
		}
$$
$$e_3e_4 d= \TikZ{[scale = .45]
		\draw[dotted] (.5,4) to  (4.5,4)  (.5,3) to  (4.5,3) (.5,2) to (4.5,2) (.5, 0) to (4.5, 0);
		\draw[line width=1pt] (.5,4) to (.5, 0) (4.5,4) to (4.5, 0);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend left] (.5,.7) node[swV]{}
			(4,2) node[sV]{} to [bend left] (.5,.3) node[swV]{};
		\begin{scope}[every node/.style={sV}]
		\draw[bend right] (4,3) node{} to  (4.5,2.7) node[swV]{}  (4.5, 2.3) node[swV]{} to (4,2) node{};
		\draw (3,2) to (3,3) node{} (2,2) to (2,3) node{} (1,2) to (1,3) node{};
		\draw[bend left=80] (3,3) to  (4,3)  (4,4) node{} to (3,4)node{};
		\draw (1,3) to (1,4) node{} (2,3) to (2,4) node{};
		\end{scope}
		}
		= 
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,0) to (4.5,0);
		\draw[line width=1pt] (.5,2) to (.5, 0) (4.5,2) to (4.5,0);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend right=80] (4,2) node[sV]{}
			(.5,.7) node[swV]{} to  (4.5,.7) node[swV]{}
			(.5,.3) node[swV]{} to  (4.5,.3) node[swV]{}
			;
		}
		=
		z
		\TikZ{[scale = .45]
		\draw[dotted] (.5,2) to  (4.5,2) (.5,0) to (4.5,0);
		\draw[line width=1pt] (.5,2) to (.5, 0) (4.5,2) to (4.5,0);
		\draw (1,2) node[sV]{} to [bend left] (.5,1.5) node[swV]{}
			(2,2) node[sV]{} to [bend left] (.5,1.1) node[swV]{}
			(3,2) node[sV]{} to [bend right=80] (4,2) node[sV]{}
			;
		}
		$$
		
		

\uncover<2->{You can tell when to use \centerline{$\displaystyle
\TikZ{[diagram]
\draw[line width=1.5pt] (.5,0)--(.5,.75);
\draw[line width=1.5pt] (2.5,0)--(2.5,.75);
\node[wV] (l1) at (.5, .2){}; \node[wV] (r1) at (2.5, .2){}; \draw (l1)--(r1);
\node[wV] (l2) at (.5, .55){}; \node[wV] (r2) at (2.5, .55){}; \draw (l2)--(r2);
\only<1-4>{\foreach\x in {.5,2.5}{\foreach\y in {-.15,.9}{\foreach \d in {-1,0,1}{
		\node[draw, fill =black, circle, inner sep=0pt, minimum size=.25pt] at (\x, \y+.1*\d) {};}}}}
} = z
$}
or not by the parity of connections to the left/right walls.}
\end{frame}


\begin{frame}
\vspace{-.25cm}
$$\hspace{-.5cm}
\TikZ{[yscale=-1.5]
\node[left] at (-1.5, 1) {\begin{tabular}{r} \bf\alert{ Generic module:}\\
\scriptsize(act by $e_i$, don't make loops)\end{tabular}};
\uncover<3->{\node[left, alert] at (-1.5, 1.5) {\begin{tabular}{r} 
	\scriptsize Red arrows indicate coef of $z$.
	\end{tabular}};}
\foreach \y in {4,5,6,7,8}{\coordinate (L\y) at (-1,\y*.55);	\coordinate (R\y) at (1,\y*.55);}
\foreach \y in {1,2,3,9,10,11}{\coordinate (C\y) at (0,\y*.55);}
\begin{scope}[every node/.style={inner sep = 1pt}]
\node (n1) at (C1) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend left] (.5,-.5)
			(4,1) to [bend left] (.5,-1);
		\draw[dotted] (.5,1) rectangle (4.5,-1.5);
		\end{tikzpicture}};
\uncover<2->{\node (n2) at (C2) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend left] (.5,-.5)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-1);
		\end{tikzpicture}};}
\uncover<3->{\node (n3) at (C3) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend right=60] (4,1);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};}
\uncover<4->{\node (nL4) at (L4) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend right] (4.5,0)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};}
\uncover<5->{\node (nR4) at (R4) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend right=60] (3,1)
			(4,1) to [bend left] (.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};}
\uncover<6->{\node (nL5) at (L5)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend right=60] (3,1)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,0);
		\end{tikzpicture}};
\node (nR5) at (R5) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend left] (.5,.5)
			(4,1) to [bend left] (.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nL6) at (L6)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend right] (4.5,0)
			(3,1) to [bend right=60] (4,1);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nR6) at (R6)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend left] (.5,0)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nL7) at (L7) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (4,1) to [bend right] (4.5,.5)
			(3,1) to [bend right] (4.5,0)
			(2,1) to [bend right] (4.5,-.5)
			(1,1) to [bend left] (.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-1);
		\end{tikzpicture}};
\node (nR7) at (R7) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend right=60] (4,1);
		\draw[dotted] (.5,1) rectangle (4.5,0);
		\end{tikzpicture}};
\node (nL8) at (L8) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend right] (4.5,0)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nR8) at (R8)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (4,1)
			(2,1) to [bend right=60] (3,1);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (n9) at (C9) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (4,1) to [bend right] (4.5,.5)
			(2,1) to [bend right=60] (3,1)
			(1,1) to [bend right] (4.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-1);
		\end{tikzpicture}};
\node (n10) at (C10) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (3,1) to [bend right=60] (4,1)
			(2,1) to [bend right] (4.5,0.5)
			(1,1) to [bend right] (4.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (n11) at (C11) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (4,1) to [bend right] (4.5,.5)
			(3,1) to [bend right] (4.5,0)
			(2,1) to [bend right] (4.5,-.5)
			(1,1) to [bend right] (4.5,-1);
		\draw[dotted] (.5,1) rectangle (4.5,-1.5);
		\end{tikzpicture}};}
\end{scope}
\uncover<2->{\draw[->] (n1) to node[right] {\tiny$e_4$} (n2);}
\uncover<3->{\draw[->, alert] (n2) to node[right] {\tiny$e_3$} (n3);}
\uncover<4->{\draw[->] (n3) to node[left] {\tiny$e_4$} (nL4);}
\uncover<5->{\draw[->] (n3) to node[right] {\tiny$e_2$} (nR4);}
\uncover<6->{\draw[->] (nL4) to node[right] {\tiny$e_2$} (nL5);
\draw[->] (nR4) to node[below right] {\tiny$e_4$} (nL5);
\draw[->] (nR4) to node[right] {\tiny$e_1$} (nR5);
\draw[->] (nL5) to node[right] {\tiny$e_3$} (nL6);
\draw[->] (nL5) to node[above right] {\tiny$e_1$} (nR6);
\draw[->] (nR5) to node[right] {\tiny$e_4$} (nR6);
\draw[->] (nL6) to node[right] {\tiny$e_4$} (nL7);
\draw[->, alert] (nL6) to node[above right] {\tiny$e_1$} (nR7);
\draw[->, alert] (nR6) to node[right] {\tiny$e_3$} (nR7);
\draw[->, alert] (nL7) to node[right] {\tiny$e_1$} (nL8);
\draw[->] (nR7) to node[below right] {\tiny$e_4$} (nL8);
\draw[->] (nR7) to node[right] {\tiny$e_2$} (nR8);
\draw[->] (nL8) to node[right] {\tiny$e_2$} (n9);
\draw[->] (nR8) to node[left] {\tiny$e_4$} (n9);
\draw[->] (n9) to node[right] {\tiny$e_3$} (n10);
\draw[->] (n10) to node[right] {\tiny$e_4$} (n11);}
}
\qquad
\uncover<7->{\begin{minipage}{2.8cm}
For what $z$ does this module split?
\end{minipage}}
%\qquad \uncover<6->{
%\TikZ{[yscale=-1.5]
%\node[left, alert] at (-1.5, 1) {\bf \begin{tabular}{l}Hecke\\algebra\\module:\end{tabular}};
%\foreach \y in {4,5,6,7,8}{\coordinate (L\y) at (-1,\y*.55);	\coordinate (R\y) at (1,\y*.55);}
%\foreach \y in {1,2,3,9,10,11}{\coordinate (C\y) at (0,\y*.55);}
%\begin{scope}[every node/.style={inner sep = 1pt}]
%\node[] (n1) at (C1) {$\Row{1,2,3,4}$};
%\node (n2) at (C2) {$\Row{-1,2,3,4}$};
%\node (n3) at (C3) {$\Row{-2,1,3,4}$};
%\node (nL4) at (L4) {$\Row{-2,-1,3,4}$};
%\node (nR4) at (R4) {$\Row{-3,1,2,4}$};
%\node (nL5) at (L5) {$\Row{-3,-1,2,4}$};
%\node (nR5) at (R5) {$\Row{-4,1,2,3}$};
%\node (nL6) at (L6) {$\Row{-3,-2,1,4}$};
%\node (nR6) at (R6) {$\Row{-4,-1,2,3}$};
%\node (nL7) at (L7) {$\Row{-3,-2,-1,4}$};
%\node (nR7) at (R7) {$\Row{-4,-2,1,3}$};
%\node (nL8) at (L8) {$\Row{-4,-2,-1,3}$};
%\node (nR8) at (R8) {$\Row{-4,-3,1,2}$};
%\node (n9) at (C9) {$\Row{-4,-3,-1,2}$};
%\node (n10) at (C10) {$\Row{-4,-3,-2,1}$};
%\node (n11) at (C11) {$\Row{-4,-3,-2,-1}$};
%\end{scope}
%\draw[<->] (n1) to node[right] {\tiny$\tau_0$} (n2);
%\draw[<->] (n2) to node[right] {\tiny$\tau_1$} (n3);
%\draw[<->] (n3) to node[right] {\tiny$\tau_0$} (nL4);
%\draw[<->] (n3) to node[right] {\tiny$\tau_2$} (nR4);
%\draw[<->] (nL4) to node[right] {\tiny$\tau_2$} (nL5);
%\draw[<->] (nR4) to node[above] {\tiny$\tau_0$} (nL5);
%\draw[<->] (nR4) to node[right] {\tiny$\tau_3$} (nR5);
%\draw[<->] (nL5) to node[right] {\tiny$\tau_1$} (nL6);
%\draw[<->] (nL5) to node[above] {\tiny$\tau_3$} (nR6);
%\draw[<->] (nR5) to node[right] {\tiny$\tau_0$} (nR6);
%\draw[<->] (nL6) to node[right] {\tiny$\tau_0$} (nL7);
%\draw[<->] (nL6) to node[above] {\tiny$\tau_3$} (nR7);
%\draw[<->] (nR6) to node[right] {\tiny$\tau_1$} (nR7);
%\draw[<->] (nL7) to node[right] {\tiny$\tau_3$} (nL8);
%\draw[<->] (nR7) to node[above] {\tiny$\tau_0$} (nL8);
%\draw[<->] (nR7) to node[right] {\tiny$\tau_2$} (nR8);
%\draw[<->] (nL8) to node[ right] {\tiny$\tau_2$} (n9);
%\draw[<->] (nR8) to node[ right] {\tiny$\tau_0$} (n9);
%\draw[<->] (n9) to node[right] {\tiny$\tau_1$} (n10);
%\draw[<->] (n10) to node[right] {\tiny$\tau_0$} (n11);
%}}
$$
\end{frame}


%
%\begin{frame}{Hecke algebras}
%{\color{black}\footnotesize For nice choices of modules and constants\dots}
%\centerline{\begin{tabular}{c@{\qquad}c}
%$TL_k^{(0)}$ & \uncover<2->{Finite type A Hecke algebra} \\
%centralizes $U_q\fsl_2$ 
%	 &\uncover<2->{centralizes $U_q\fsl_n$}\\
%on $L(1)^{\otimes k}$ & \uncover<2->{on $L(\omega_1)^{\otimes k}$}\\~\\
%$TL_k^{(1)}$ & \uncover<2->{Finite type B Hecke algebra} \\
%centralizes $U_q\fsl_2$ 
%	 &\uncover<2->{centralizes $U_q\fsl_n$}\\
%on $L(\ell) \otimes L(1)^{\otimes k}$ & \uncover<2->{on $L(\lambda) \otimes L(\omega_1)^{\otimes k}$}\\~\\
%$TL_k^{(2)}$ & \uncover<2->{\only<5>{\color{black!20}}Affine type C Hecke algebra} \\
%centralizes $U_q\fsl_2$ 
%	 &\uncover<2->{\only<5>{\color{black!20}}centralizes $U_q\fsl_n$}\\
%on $L(\ell) \otimes L(1)^{\otimes k} \otimes L(m)$ & \uncover<2->{\only<5>{\color{black!20}}on $L(\lambda) \otimes L(\omega_1)^{\otimes k}\otimes L(\mu)$}\\~\\
%\uncover<4->{\alert<4>{$\uparrow$}}	&\uncover<3->{\alert<3>{$\uparrow$}}\\
%\uncover<4->{\alert<4>{Quotients of}}	&\uncover<3->{\alert<3>{Hecke reps organized by}}\\
%\uncover<4->{\alert<4>{corresponding Hecke algebras}}	&\uncover<3->{\alert<3>{weight spaces for Murphy elements,}}\\
%	&\uncover<3->{\alert<3>{actions written with intertwiners.}}\\
%\end{tabular}}
%\end{frame}



\begin{frame}
\hbox{\hspace{-.4in}\includegraphics{diagrams-slideTL}}
\end{frame}

\begin{frame}%TABLE OF DIAGRAM ALGEBRAS TBdry
\hbox{\hspace{-.4in}\includegraphics{diagrams-slide-TBdry}}
\end{frame}




%%%%%%%%%%%%%%%


\begin{frame}
\def\SCALE{.6}
The \GREEN{two-boundary (two-pole) braid group} $\cB_k$ is generated by 
\centerline{
\def\TOP{1}\def\K{1}
$
T_k=\TikZ{[diagram]
\Under[\K,0][\K+.8,.5]
\Pole[1.5][0,1]
\Over[\K+.8,.5][\K,1]
\Caps[1.5][0,1][1]
	}, \quad 
T_0= \TikZ{[diagram]
	\Under[.2,.5][1,1]
	\Pole[.5][0,1]
	\Over[1,0][.2,.5]
	\Caps[.5][0,1][1]
	}\quad \text{and} \quad 
T_i=\TikZ{[diagram]
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[0,1][1][\tiny $i$]
	\Label[0,1][2][\tiny$i$+1]
}
\qquad \text{for }1\leq i \leq k-1$,}
\pause
subject to relations\\
\uncover<5->{
\centerline{\begin{tikzpicture}[line width=.75pt]
	\foreach \x in {0,1,2,4,5,6}{
	\node[draw, thick, fill=white, circle, inner sep=0pt, minimum size=8pt] (\x) at (\x,0){};
	\ifnum \x<3 \node[above, outer sep=3pt] at (\x) {\small $T_\x$}; \fi
	}
	\node[above, outer sep=3pt] at (4) {\small$T_{k-2}$}; 
		\node[above, outer sep=3pt] at (5) {\small$T_{k-1}$}; 
		\node[above, outer sep=3pt] at (6) {\small$T_{k}$};
	\draw[double, double distance=3pt] (0) to (1) (5) to (6);
	\draw(1) to (2) (4) to (5);
	\draw[dashed] (4) to (2);
\end{tikzpicture}}
i.e.\ }


\centerline{$
\def\TOP{2.5}
T_i T_{i+1} T_i ~=
\TikZ{[diagram]
	\Under[2,1.5][1,.75]
	\Under[3,1.2][1,0]
	\Over[1,1.5][3,.3]
	\Over[1,.75][2,0]
	\draw(3,1.5)--(3,1.2) (3,.3)--(3,0);
	\Nodes[0][3]
	\Nodes[1.5][3]
}
=
\TikZ{[diagram]
	\Under[3,.75][2,0]
	\Under[3,1.5][1,.3]
	\Over[1,1.2][3,0]
	\Over[2,1.5][3,.75]
	\draw(1,1.5)--(1,1.2) (1,.3)--(1,0);
	\Nodes[0][3]
	\Nodes[1.5][3]
}
= ~T_{i+1} T_i T_{i+1},
$}
\pause
\centerline{$\def\TOP{2.5}
T_1 T_0 T_1 T_0~=
\TikZ{[diagram]
	\Under[.2,.5][2,1.5]
	\Under[0,1.5][2,2.5]
	\Pole[.5][0,\TOP]
	\Over[0,1.5][2,.7]
	\Over[1,0][.2,.5]
	\draw (2,.7)--(2,0);
	\Over[1,2.5][2,1.7]
	\draw (2,1.7)--(2,1.5);
	\Caps[.5][0,\TOP][2]
}
=
\TikZ{[diagram]
	\Under[1,2.5][.2,2]
	\Under[2,2][0,1]
	\Under[2,.7][1,0]
	\Pole[.5][0,\TOP]
	\Over[.2,2][2,1.2]
	\Over[0,1][2,0]
	\draw (2,2.5)--(2,2);
	\draw (2,1.2)--(2,.7);
	\Caps[.5][0,\TOP][2]
}
= ~ T_0 T_1 T_0 T_1,$}\pause
\centerline{and, similarly, $T_{k-1}T_{k}T_{k-1}T_{k}=T_{k}T_{k-1}T_{k}T_{k-1}$.}
\end{frame}





%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}\small
\def\SCALE{.5}
\GREEN{(1)} The \GREEN{two-boundary (two-pole) braid group} $\cB_k$ is generated by 
\centerline{
\def\TOP{1}\def\K{1}
$
T_k=\TikZ{[diagram]
\Under[\K,0][\K+.8,.5]
\Pole[1.5][0,1]
\Over[\K+.8,.5][\K,1]
\Caps[1.5][0,1][1]
}, \quad 
T_0= \TikZ{[diagram]
	\Under[.2,.5][1,1]
	\Pole[.5][0,1]
	\Over[1,0][.2,.5]
	\Caps[.5][0,1][1]
}\quad \text{and} \quad T_i=\TikZ{[diagram]
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[0,1][1][\tiny $i$]
	\Label[0,1][2][\tiny$i$+1]
}
\qquad \text{for }1\leq i \leq k-1$,}
subject to relations 
$\TikZ{[line width=.75pt, scale=.75]
	\foreach \x in {0,1,2,4,5,6}{
	\node[draw, thick, fill=white, circle, inner sep=0pt, minimum size=8pt] (\x) at (\x,0){};
	\ifnum \x<3 \node[above, outer sep=3pt] at (\x) {\scriptsize $T_\x$}; \fi
	}
	\node[above, outer sep=3pt] at (4) {\scriptsize$T_{k-2}$}; 
		\node[above, outer sep=3pt] at (5) {\scriptsize$T_{k-1}$}; 
		\node[above, outer sep=3pt] at (6) {\scriptsize$T_{k}$};
	\draw[double, double distance=3pt] (0) to (1) (5) to (6);
	\draw(1) to (2) (4) to (5);
	\draw[dashed] (4) to (2);
}$.

\medskip

\uncover<2->{
{\color{TypeA}(2)} Fix  constants $t_0, t_{k},t\only<3->{=t_1=t_2=\cdots=t_{k-1}} \in \CC$. \\
The {\color{TypeA}affine type C Hecke algebra}  $\cH_k$ is the quotient of $\CC\cB_k$ by the relations \only<3->{$(T_i - t_i^{1/2})(T_i + t_i^{-1/2}) = 0$.}\only<1-2>{$$(T_0 - t_0^{1/2})(T_0 + t_0^{-1/2})=0, \quad (T_k - t_k^{1/2})(T_k + t_k^{-1/2})=0$$
$$\text{and } \quad (T_i - t^{1/2})(T_i + t^{-1/2}) = 0 \quad \text{for $i=1, \dots, k-1$.}$$}}

\medskip

\uncover<4->{
{\color{TypeA2}(3)} Set
\begin{align*}
\TikZ{[diagram]
	\draw[line width=1.5pt, TypeA2] (.5,0) to (.5,1.25);
	\draw[densely dotted, TypeA2] (.5,0) to (1.5,0) (.5,1.25) to (1.5,1.25);	
	\node[V] (b1) at (1, 0){};	
	\node[V] (t1) at (1, 1.25) {};
	\node[wV] (L1) at (.5,.45) {}; 
	\node[wV] (L2) at (.5,.8) {}; 
	\draw[bend right=50] (L2) to (t1) (b1) to (L1); 
	}
	&\ =\     t_0^{1/2}\ \TikZ{[diagram]
			\Pole[.5][0,1]
			\draw (1,0) to (1,1);
			\Caps[.5][0,1][1]
		}
		\ -\  \TikZ{[diagram]
			\Under[.2,.5][1,1]
			\Pole[.5][0,1]
			\Over[1,0][.2,.5]
			\Caps[.5][0,1][1]
		}
		& (e_0 =t_0^{1/2} - T_0)\\
\TikZ{[diagram]
	\draw[line width=1.5pt, TypeA2] (1.5,0) to (1.5,1.25);
	\draw[densely dotted, TypeA2] (.5,0) to (1.5,0) (.5,1.25) to (1.5,1.25);	
	\node[V] (b1) at (1, 0){};	
	\node[V] (t1) at (1, 1.25) {};
	\node[wV] (L1) at (1.5,.45) {}; 
	\node[wV] (L2) at (1.5,.8) {}; 
	\draw[bend left=50] (L2) to (t1) (b1) to (L1); 
	}
	&\ =\     t_k^{1/2}\ \TikZ{[diagram]
			\Pole[1.5][0,1]
			\draw (1,0) to (1,1);
			\Caps[1.5][0,1][1]
		}
		\ - \ \TikZ{[diagram]
			\Under[1,0][1+.8,.5]
			\Pole[1.5][0,1]
			\Over[1+.8,.5][1,1]
			\Caps[1.5][0,1][1]
			}
		& (e_k =t_k^{1/2} - T_k)\\
\TikZ{[diagram]
	\draw[densely dotted, TypeA2] (.75,0) to (2,0) (.75,1) to (2,1);	
	\node[V] (b1) at (1, 0){};	\node[V] (b2) at (1.75, 0){};	
	\node[V] (t1) at (1, 1) {}; \node[V] (t2) at (1.75, 1){};	
	\draw[bend left=60] (t2) to (t1) (b1) to (b2); 
	}
	&\ = \ t^{1/2} \ \TikZ{[diagram]
			\node[V] (b1) at (1, 0){};	\node[V] (b2) at (1.75, 0){};	
			\node[V] (t1) at (1, 1) {}; \node[V] (t2) at (1.75, 1){};	
			\draw(b1) to (t1) (b2) to (t2);
			}
	\ - \ \TikZ{[diagram]
			\Cross[1,0][1.75,1]
			\node[V] (b1) at (1, 0){};	\node[V] (b2) at (1.75, 0){};	
			\node[V] (t1) at (1, 1) {}; \node[V] (t2) at (1.75, 1){};	
			}
		& (e_i =t^{1/2} - T_i)
\end{align*}\vspace{-.1in}

so that $e_j^2 = z_j e_j$ (for good $z_j$).}\\
\uncover<5->{The {\color{TypeA2}two-boundary Temperley-Lieb algebra} is the quotient of $\cH_k$ by the relations 
$e_i e_{i\pm 1} e_i = e_i$ for $i = 1, \dots, k-1$.}
\end{frame}





%%%%%%%%%%%%%
%%%%%%%%%%%%%
\begin{frame}\small
\def\SCALE{.5}
\GREEN{(1)} The \GREEN{two-boundary (two-pole) braid group} $\cB_k$ is generated by 
\centerline{
\def\TOP{1}\def\K{1}
$
T_k=\TikZ{[diagram]
\Under[\K,0][\K+.8,.5]
\Pole[1.5][0,1]
\Over[\K+.8,.5][\K,1]
\Caps[1.5][0,1][1]
}, \quad 
T_0= \TikZ{[diagram]
	\Under[.2,.5][1,1]
	\Pole[.5][0,1]
	\Over[1,0][.2,.5]
	\Caps[.5][0,1][1]
}\quad \text{and} \quad T_i=\TikZ{[diagram]
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[0,1][1][\tiny $i$]
	\Label[0,1][2][\tiny$i$+1]
}
\qquad \text{for }1\leq i \leq k-1$.}

\medskip


{\color{TypeA}(2)} Fix  constants $t_0, t_{k},t=t_1=t_2=\cdots=t_{k-1} \in \CC$. \\
The {\color{TypeA}affine type C Hecke algebra}  $\cH_k$ is the quotient of $\CC\cB_k$ by the relations $(T_i - t_i^{1/2})(T_i + t_i^{-1/2}) = 0$.

\medskip


{\color{TypeA2}(3)} Set
$$
\TikZ{[diagram]
	\draw[line width=1.5pt, TypeA2] (.5,0) to (.5,1.25);
	\draw[densely dotted, TypeA2] (.5,0) to (1.5,0) (.5,1.25) to (1.5,1.25);	
	\node[V] (b1) at (1, 0){};	
	\node[V] (t1) at (1, 1.25) {};
	\node[wV] (L1) at (.5,.45) {}; 
	\node[wV] (L2) at (.5,.8) {}; 
	\draw[bend right=50] (L2) to (t1) (b1) to (L1); 
	}
	=    t_0^{1/2}\TikZ{[diagram]
			\Pole[.5][0,1]
			\draw (1,0) to (1,1);
			\Caps[.5][0,1][1]
		}
		- \TikZ{[diagram]
			\Under[.2,.5][1,1]
			\Pole[.5][0,1]
			\Over[1,0][.2,.5]
			\Caps[.5][0,1][1]
		}, \quad
\TikZ{[diagram]
	\draw[line width=1.5pt, TypeA2] (1.5,0) to (1.5,1.25);
	\draw[densely dotted, TypeA2] (.5,0) to (1.5,0) (.5,1.25) to (1.5,1.25);	
	\node[V] (b1) at (1, 0){};	
	\node[V] (t1) at (1, 1.25) {};
	\node[wV] (L1) at (1.5,.45) {}; 
	\node[wV] (L2) at (1.5,.8) {}; 
	\draw[bend left=50] (L2) to (t1) (b1) to (L1); 
	}
	=    t_k^{1/2}\TikZ{[diagram]
			\Pole[1.5][0,1]
			\draw (1,0) to (1,1);
			\Caps[1.5][0,1][1]
		}
		- \TikZ{[diagram]
			\Under[1,0][1+.8,.5]
			\Pole[1.5][0,1]
			\Over[1+.8,.5][1,1]
			\Caps[1.5][0,1][1]
			} \quad \text{and} \quad 
	\TikZ{[diagram]
	\draw[densely dotted, TypeA2] (.75,0) to (2,0) (.75,1) to (2,1);	
	\node[V] (b1) at (1, 0){};	\node[V] (b2) at (1.75, 0){};	
	\node[V] (t1) at (1, 1) {}; \node[V] (t2) at (1.75, 1){};	
	\draw[bend left=60] (t2) to (t1) (b1) to (b2); 
	}
	= t^{1/2} 
		\TikZ{[diagram]
			\node[V] (b1) at (1, 0){};	\node[V] (b2) at (1.75, 0){};	
			\node[V] (t1) at (1, 1) {}; \node[V] (t2) at (1.75, 1){};	
			\draw(b1) to (t1) (b2) to (t2);
			}
	- 
		\TikZ{[diagram]
			\Cross[1,0][1.75,1]
			\node[V] (b1) at (1, 0){};	\node[V] (b2) at (1.75, 0){};	
			\node[V] (t1) at (1, 1) {}; \node[V] (t2) at (1.75, 1){};	
			}$$
so that $e_j^2 = z_j e_j$. The {\color{TypeA2}two-boundary Temperley-Lieb algebra} is the quotient of $\cH_k$ by the relations 
$e_i e_{i\pm 1} e_i = e_i$ for $i = 1, \dots, k-1$.

\medskip

\uncover<2->{\centerline{\includegraphics{diagrams-slide-FR}}
\centerline{\includegraphics{diagrams-slide-LR}}}

\end{frame}



\begin{frame}{Representation theory of $\cH_k$}\small

The representations of $\cH_k$ are indexed by pairs $(\cc, J)$, where 
\begin{align*}
	\cc & \text{ is a point in the fundamental chamber of}\\
		& \text{ the (finite) type C hyperplane system, and }\\
	J & \text{ is a set of choices of positive/negative sides of}\\
		& \text{ other distinguished hyperplanes intersecting $\cc$}
\end{align*}

\pause


\centerline{Example: $k=2$ \quad 
$\TikZ{[scale=.7]
\uncover<3->{\filldraw[gray!30] (0,3)--(0,0)--(3,3)--(0,3);}
\uncover<3->{\filldraw[red] (1.5,2.5) circle (2pt) node[above]{\tiny$(c_1, c_2)$};}
\draw[thick] (0,-3)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-3,0)--(3,0) node[right] {$\fh^{\alpha_2+2\alpha_1}$};
\draw[thick] (3,-3)--(-3,3) node[above] {$\fh^{\alpha_1+\alpha_2}$};
\draw[thick] (-3,-3)--(3,3) node[above right] {$\fh^{\alpha_1}$};
\uncover<4->{
	\draw[dotted, thick] (3-.4,3)--(-3,-3+.4) node[left]{\tiny$c_2\!=\!c_1\!+\!1$};
	\draw[dotted, thick] (-3+.4,3)--(3,-3+.4) node[right]{\tiny$c_2\!=\!-\!c_1\!+\!1$};
	\draw[dotted, thick] (-3,3-.4)--(3-.4,-3) node[below right]{\tiny$c_2\!=\!-\!c_1\!-\!1$};
	\draw[dotted, thick] (3,3-.4)--(-3+.4,-3) node[below left]{\tiny$c_2\!=\!c_1\!-\!1$};
	}
\uncover<5->{
	\draw[dotted, thick] (3,.8)--(-3,.8) node[left] {\tiny$c_2\!=\!r_1$};
	\draw[dotted, thick] (3,2.2)--(-3,2.2) node[left] {\tiny$c_2\!=\!r_2$};
	\draw[dotted, thick] (.8,-3)--(.8,3) node[above] {\tiny$c_1\!=\!r_1$};
	\draw[dotted, thick] (2.2,-3)--(2.2,3) node[above] {\tiny$c_1\!=\!r_2$};
	\draw[dotted, thick] (3,-.8)--(-3,-.8) node[left] {\tiny$c_2\!=\!-\!r_1$};
	\draw[dotted, thick] (3,-2.2)--(-3,-2.2) node[left] {\tiny$c_2\!=\!-\!r_2$};
	\draw[dotted, thick] (-.8,3)--(-.8,-3.2) node[below] {\tiny$c_1\!=\!-\!r_1$};
	\draw[dotted, thick] (-2.2,3)--(-2.2,-3.2) node[below] {\tiny$c_1\!=\!-\!r_2$};
}
}$
}
\uncover<5->{\vspace{-.09in} \scriptsize  \uncover<5->{The $r_i$s depend on $\cH_k$'s parameters $t_0$ and $t_k$: $r_1 = \log_t(t_0/t_k)$, $r_2 = \log_t(t_0t_k)$}}
\end{frame}


\begin{frame}{Representation theory of $\cH_k$}

\centerline{
\begin{tikzpicture}[scale=1.875]
	\filldraw[gray!20] (0,3)--(0,0)--(3,3)--(0,3);
\draw[thick] (0,-.2)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-.2,-.2)--(3,3) node[above right] {$\fh^{\alpha_1}$};
\draw[gray] (-.2,0)--(3,0) node[right] {\color{gray}$\fh^{\alpha_2+2\alpha_1}$};
	\draw[dotted, thick, black!70] (3-.4,3)--(-.2,-.2+.4) node[left]{\tiny$c_2=c_1+1$};
	\draw[dotted, thick, black!70] (-.2,.2+.4)--(.4+.2,-.2) node[right]{\tiny$c_2=-c_1+1$};
	\draw[dotted, thick, black!70] (3,.6)--(-.2,.6) node[left] {\tiny$c_2=r_1$};
	\draw[dotted, thick, black!70] (3,2.4)--(-.2,2.4) node[left] {\tiny$c_2=r_2$};
	\draw[dotted, thick, black!70] (.6,-.2)--(.6,3) node[above] {\tiny$c_1=r_1$};
	\draw[dotted, thick, black!70] (2.4,-.2)--(2.4,3) node[above] {\tiny$c_1=r_2$};
\uncover<4->{
	\filldraw[red!50] (1.3+.2,2.4)--(1.3+.15,2.4+.15)--(1.3,2.4+.2)--(1.3-.15,2.4+.15)--(1.3-.2,2.4);
			\filldraw[yellow!70] (1.3+.2,2.4)--(1.3+.15,2.4-.15)
							--(1.3,2.4-.2)--(1.3-.15,2.4-.15)--(1.3-.2,2.4);}
			\uncover<3->{\REG[1.3,2.4]}
\uncover<8->{	\filldraw[blue!50] (1+.2,2)--(1+.15,2+.15)--(1,2+.2)
					--(1-.15,2+.15)--(1-.2,2)--(1-.15,2-.15)
					--(1,2-.2)--(1+.15,2-.15); 
					\REG[1,2]}
\uncover<6->{	\filldraw[red!50] (.6+.15,1+.15)--(.6,1+.2)--(.6,1); 
			\filldraw[yellow!70] (.6,1+.2)--(.6-.15,1+.15)--(.6-.2,1)--(.6-.15,1-.15)--(.6,1);
			\filldraw[blue!50] (.6-.15,1-.15)--(.6,1-.2)--(.6,1);
			\filldraw[green!50](.6,1-.2)--(.6+.15,1-.15)--(.6+.2,1)--(.6+.15,1+.15)--(.6,1); }
			\uncover<5->{\REG[.6,1]}
\uncover<7->{	\filldraw[green!50] (.6,2.4)--(.6,2.4+.2)--(.6-.15,2.4+.15)--(.6-.2,2.4);
		\filldraw[yellow!70] (.6,2.4)--(.6-.2,2.4)--(.6-.15,2.4-.15)--(.6,2.4-.2);
		\filldraw[blue!50] (.6,2.4)--(.6,2.4-.2)--(.6+.15,2.4-.15)--(.6+.2,2.4);
		\filldraw[red!50] (.6,2.4)--(.6+.2,2.4)--(.6+.15,2.4+.15)--(.6,2.4+.2);
		\REG[.6,2.4]}
%\uncover<9->{%	\filldraw[red!50] (2.4,2.4)--(2.4-.2,2.4)--(2.4-.15,2.4+.15)--(2.4,2.4+.2)--(2.4,2.4);
%		\REGnw[2.4,2.4]
%		\REGe[0,1.7]}
\uncover<2->{
	% 2
	\filldraw[red] (.6,2.4) circle (1pt) node[above]{};
	% 3/1
	\filldraw[red] (.2,.6) circle (1pt) node[above]{};
	\filldraw[red] (.6,1) circle (1pt) node[above]{};
	\filldraw[red] (2,2.4) circle (1pt) node[above]{};
	\filldraw[red] (2.4,2.8) circle (1pt) node[above]{};
	%4:
	\filldraw[red] (.6,1.7) circle (1pt) node[above]{};
	\filldraw[red] (1.3,1.7) circle (1pt) node[above]{};
	\filldraw[red] (1.3,2.4) circle (1pt) node[above]{};
	\filldraw[red] (.1,.3) circle (1pt) node[above]{};
	\filldraw[red] (.4,.6) circle (1pt) node[above]{};
	%8
	\filldraw[red] (1,2) circle (1pt) node[above]{};
%	%Walls
%	\filldraw[blue] (0,1.7) circle (1pt) node[above]{};
%	\filldraw[blue] (2.4,2.4) circle (1pt) node[above]{};
	}
\end{tikzpicture}}
{ \scriptsize The $r_i$s depend on $\cH_k$'s parameters $t_0$ and $t_k$: $r_1 = \log_t(t_0/t_k)$, $r_2 = \log_t(t_0t_k)$}
\end{frame}





%%%%%%%%%%
%%%%%%%%%%
\begin{frame}
\centerline{$\TikZ{[scale=1.875]
	\filldraw[gray!20] (0,3)--(0,0)--(3,3)--(0,3);
\draw[thick] (0,-.2)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-.2,-.2)--(3,3) node[above right] {$\fh^{\alpha_1}$};
\draw[gray] (-.2,0)--(3,0) node[right] {\color{gray}$\fh^{\alpha_2+2\alpha_1}$};
	\draw[dotted, thick, black!70] (3-.4,3)--(-.2,-.2+.4) node[left]{\tiny$c_2=c_1+1$};
	\draw[dotted, thick, black!70] (-.2,.2+.4)--(.4+.2,-.2) node[right]{\tiny$c_2=-c_1+1$};
	\draw[dotted, thick, black!70] (3,.6)--(-.2,.6) node[left] {\tiny$c_2=r_1$};
	\draw[dotted, thick, black!70] (3,2.4)--(-.2,2.4) node[left] {\tiny$c_2=r_2$};
	\draw[dotted, thick, black!70] (.6,-.2)--(.6,3) node[above] {\tiny$c_1=r_1$};
	\draw[dotted, thick, black!70] (2.4,-.2)--(2.4,3) node[above] {\tiny$c_1=r_2$};
\uncover<2->{	
	\draw[red, very thick] (3-.4,3)--(0,.4)--(.2,.2);
		}
\uncover<1,3->{
	\uncover<1>{
		\filldraw[red!50] (1.3+.2,2.4)--(1.3+.15,2.4+.15)--(1.3,2.4+.2)--(1.3-.15,2.4+.15)--(1.3-.2,2.4);
			\filldraw[yellow!70] (1.3+.2,2.4)--(1.3+.15,2.4-.15)
							--(1.3,2.4-.2)--(1.3-.15,2.4-.15)--(1.3-.2,2.4);
			\REG[1.3,2.4]
		%
		\filldraw[blue!50] (1+.2,2)--(1+.15,2+.15)--(1,2+.2)
					--(1-.15,2+.15)--(1-.2,2)--(1-.15,2-.15)
					--(1,2-.2)--(1+.15,2-.15); 
					\REG[1,2]
	%r_1,r_2:
	\filldraw[green!50] (.6,2.4)--(.6,2.4+.2)--(.6-.15,2.4+.15)--(.6-.2,2.4);
		\filldraw[blue!50] (.6,2.4)--(.6,2.4-.2)--(.6+.15,2.4-.15)--(.6+.2,2.4);
		\filldraw[yellow!70] (.6,2.4)--(.6-.2,2.4)--(.6-.15,2.4-.15)--(.6,2.4-.2);
				\filldraw[red!50] (.6,2.4)--(.6+.2,2.4)--(.6+.15,2.4+.15)--(.6,2.4+.2);
		\REG[.6,2.4]
		\filldraw[red] (.6,2.4) circle (1pt) node[above]{};
			%
					}
	\filldraw[red!50] (.6+.15,1+.15)--(.6,1+.2)--(.6,1); 
			\filldraw[yellow!70] (.6,1+.2)--(.6-.15,1+.15)--(.6-.2,1)--(.6-.15,1-.15)--(.6,1);
			%
			\uncover<1>{\filldraw[blue!50] (.6-.15,1-.15)--(.6,1-.2)--(.6,1);
			\filldraw[green!50](.6,1-.2)--(.6+.15,1-.15)--(.6+.2,1)--(.6+.15,1+.15)--(.6,1); }
			\REG[.6,1]
		%
\uncover<2->{	 
		\filldraw[yellow!70] (1.3+.15,1.7+.15)--(1.3,1.7+.2)--(1.3-.15,1.7+.15)--(1.3-.2,1.7)--(1.3-.15,1.7-.15)--(1.3,1.7);
			\REG[1.3,1.7]
		\filldraw[red!50] (2-.15,2.4-.15)--(2-.2,2.4)--(2,2.4); 
			\filldraw[yellow!70] (2-.2,2.4)--(2-.15,2.4+.15)--(2,2.4+.2)--(2+.15,2.4+.15)--(2,2.4);
			\REG[2,2.4]
		\filldraw[red!50] (2.4+.15,2.8+.15)--(2.4,2.8+.2)--(2.4,2.8); 
			\filldraw[yellow!70] (2.4,2.8+.2)--(2.4-.15,2.8+.15)--(2.4-.2,2.8)--(2.4-.15,2.8-.15)--(2.4,2.8);
			\REG[2.4,2.8]
		\filldraw[red!50] (0.2-0.1,0.6-0.1)--(0.2-.13,0.6)--(0.2,0.6); 
			\filldraw[yellow!70] (0.2-.13,0.6)--(0.2-0.1,0.6+0.1)--(0.2,0.6+.13)--(0.2+0.1,0.6+0.1)--(0.2,0.6);
			\sREG[0.2,0.6]
			}
\uncover<3->{
		\filldraw[green!50] (0.1-.1,0.3+.1)--(0.1,0.3+0.13)--(0.1+.1,0.3+.1)--(0.1+0.13,0.3)--(0.1+.1,0.3-.1)--(0.1,0.3);
			\sREG[.1,.3]}	
	}
%\uncover<1>{%	\filldraw[red!50] (2.4,2.4)--(2.4-.2,2.4)--(2.4-.15,2.4+.15)--(2.4,2.4+.2)--(2.4,2.4);
%		\REGnw[2.4,2.4]
%		\REGe[0,1.7]}
{
\uncover<1,3->{	% 3/1
	\filldraw[red] (.2,.6) circle (1pt) node[above]{};
	\filldraw[red] (.6,1) circle (1pt) node[above]{};
	\filldraw[red] (2,2.4) circle (1pt) node[above]{};
	\filldraw[red] (2.4,2.8) circle (1pt) node[above]{};
	%4:
	\filldraw[red] (1.3,1.7) circle (1pt) node[above]{};
	\filldraw[red] (.1,.3) circle (1pt) node[above]{};}
	\uncover<1>{
		\filldraw[red] (.6,1.7) circle (1pt) node[above]{};
		\filldraw[red] (1.3,2.4) circle (1pt) node[above]{};
			\filldraw[red] (.4,.6) circle (1pt) node[above]{};}
	%8
	\uncover<1>{\filldraw[red] (1,2) circle (1pt) node[above]{};}
	%Walls
%	\uncover<1>{
%		\filldraw[blue] (0,1.7) circle (1pt) node[above]{};
%		\filldraw[blue] (2.4,2.4) circle (1pt) node[above]{};}
	}
}$
}
\only<1-3>{
\alert{Thm. (D.-Ram)}\\
\alert{(1)} Representations of $\cH_k$ are indexed by pairs $(\cc, J)$. \\
\uncover<2->{\alert{(2)} The (calibrated) representations of $\cH_k$ that factor through the Temperley-Lieb quotient are (see above).}}
\end{frame}


\begin{frame}
\vspace{-.25cm}
$$\hspace{-.75cm}
\TikZ{[yscale=-1.5]
\node[left] at (-.6, .5) {\footnotesize \alert{Diagrams:}};
\foreach \y in {4,5,6,7,8}{\coordinate (L\y) at (-1,\y*.55);	\coordinate (R\y) at (1,\y*.55);}
\foreach \y in {1,2,3,9,10,11}{\coordinate (C\y) at (0,\y*.55);}
\begin{scope}[every node/.style={inner sep = 1pt}]
\node (n1) at (C1) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend left] (.5,-.5)
			(4,1) to [bend left] (.5,-1);
		\draw[dotted] (.5,1) rectangle (4.5,-1.5);
		\end{tikzpicture}};
\node (n2) at (C2) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend left] (.5,-.5)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-1);
		\end{tikzpicture}};
\node (n3) at (C3) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend right=60] (4,1);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nL4) at (L4) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend left] (.5,0)
			(3,1) to [bend right] (4.5,0)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nR4) at (R4) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend right=60] (3,1)
			(4,1) to [bend left] (.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nL5) at (L5)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend right=60] (3,1)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,0);
		\end{tikzpicture}};
\node (nR5) at (R5) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend left] (.5,.5)
			(4,1) to [bend left] (.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nL6) at (L6)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend left] (.5,.5)
			(2,1) to [bend right] (4.5,0)
			(3,1) to [bend right=60] (4,1);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nR6) at (R6)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend left] (.5,0)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nL7) at (L7) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (4,1) to [bend right] (4.5,.5)
			(3,1) to [bend right] (4.5,0)
			(2,1) to [bend right] (4.5,-.5)
			(1,1) to [bend left] (.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-1);
		\end{tikzpicture}};
\node (nR7) at (R7) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend right=60] (4,1);
		\draw[dotted] (.5,1) rectangle (4.5,0);
		\end{tikzpicture}};
\node (nL8) at (L8) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (2,1)
			(3,1) to [bend right] (4.5,0)
			(4,1) to [bend right] (4.5,.5);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (nR8) at (R8)  {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (1,1) to [bend right=60] (4,1)
			(2,1) to [bend right=60] (3,1);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (n9) at (C9) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (4,1) to [bend right] (4.5,.5)
			(2,1) to [bend right=60] (3,1)
			(1,1) to [bend right] (4.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-1);
		\end{tikzpicture}};
\node (n10) at (C10) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (3,1) to [bend right=60] (4,1)
			(2,1) to [bend right] (4.5,0.5)
			(1,1) to [bend right] (4.5,0);
		\draw[dotted] (.5,1) rectangle (4.5,-.5);
		\end{tikzpicture}};
\node (n11) at (C11) {\begin{tikzpicture}[xscale=.25, yscale=.2]
		\draw (4,1) to [bend right] (4.5,.5)
			(3,1) to [bend right] (4.5,0)
			(2,1) to [bend right] (4.5,-.5)
			(1,1) to [bend right] (4.5,-1);
		\draw[dotted] (.5,1) rectangle (4.5,-1.5);
		\end{tikzpicture}};
\end{scope}
\draw[->] (n1) to node[right] {\tiny$e_4$} (n2);
\draw[->, alert] (n2) to node[right] {\tiny$e_3$} (n3);
\draw[->] (n3) to node[left] {\tiny$e_4$} (nL4);
\draw[->] (n3) to node[right] {\tiny$e_2$} (nR4);
\draw[->] (nL4) to node[right] {\tiny$e_2$} (nL5);
\draw[->] (nR4) to node[below right] {\tiny$e_4$} (nL5);
\draw[->] (nR4) to node[right] {\tiny$e_1$} (nR5);
\draw[->] (nL5) to node[right] {\tiny$e_3$} (nL6);
\draw[->] (nL5) to node[above right] {\tiny$e_1$} (nR6);
\draw[->] (nR5) to node[right] {\tiny$e_4$} (nR6);
\draw[->] (nL6) to node[right] {\tiny$e_4$} (nL7);
\draw[->, alert] (nL6) to node[above right] {\tiny$e_1$} (nR7);
\draw[->, alert] (nR6) to node[right] {\tiny$e_3$} (nR7);
\draw[->, alert] (nL7) to node[right] {\tiny$e_1$} (nL8);
\draw[->] (nR7) to node[below right] {\tiny$e_4$} (nL8);
\draw[->] (nR7) to node[right] {\tiny$e_2$} (nR8);
\draw[->] (nL8) to node[right] {\tiny$e_2$} (n9);
\draw[->] (nR8) to node[left] {\tiny$e_4$} (n9);
\draw[->] (n9) to node[right] {\tiny$e_3$} (n10);
\draw[->] (n10) to node[right] {\tiny$e_4$} (n11);}
\TikZ{[yscale=-1.5]
\draw[dashed] (0,0) to (0,11*.55);
\node[left] at (0,1) {\footnotesize [GN]}; \node[right] at (0,1) {\footnotesize [DR]}; }
%%%%%%
\hspace{-.75cm}
\TikZ{[yscale=-1.5]
\node[left, alert] at (-.4, .5) { \footnotesize \begin{tabular}{r}Aff.\ type \\$C$ Hecke:\end{tabular}};
\foreach \y in {4,5,6,7,8}{\coordinate (L\y) at (-1,\y*.55);	\coordinate (R\y) at (1,\y*.55);}
\foreach \y in {1,2,3,9,10,11}{\coordinate (C\y) at (0,\y*.55);}
\begin{scope}[every node/.style={inner sep = 1pt}]
\node[] (n1) at (C1) {$\Row{1,2,3,4}$};
\node (n2) at (C2) {$\Row{-1,2,3,4}$};
\node (n3) at (C3) {$\Row{-2,1,3,4}$};
\node (nL4) at (L4) {$\Row{-2,-1,3,4}$};
\node (nR4) at (R4) {$\Row{-3,1,2,4}$};
\node (nL5) at (L5) {$\Row{-3,-1,2,4}$};
\node (nR5) at (R5) {$\Row{-4,1,2,3}$};
\node (nL6) at (L6) {$\Row{-3,-2,1,4}$};
\node (nR6) at (R6) {$\Row{-4,-1,2,3}$};
\node (nL7) at (L7) {$\Row{-3,-2,-1,4}$};
\node (nR7) at (R7) {$\Row{-4,-2,1,3}$};
\node (nL8) at (L8) {$\Row{-4,-2,-1,3}$};
\node (nR8) at (R8) {$\Row{-4,-3,1,2}$};
\node (n9) at (C9) {$\Row{-4,-3,-1,2}$};
\node (n10) at (C10) {$\Row{-4,-3,-2,1}$};
\node (n11) at (C11) {$\Row{-4,-3,-2,-1}$};
\end{scope}
\draw[<->] (n1) to node[right] {\tiny$\tau_0$} (n2);
\draw[<->] (n2) to node[right] {\tiny$\tau_1$} (n3);
\draw[<->] (n3) to node[right] {\tiny$\tau_0$} (nL4);
\draw[<->] (n3) to node[right] {\tiny$\tau_2$} (nR4);
\draw[<->] (nL4) to node[right] {\tiny$\tau_2$} (nL5);
\draw[<->] (nR4) to node[above] {\tiny$\tau_0$} (nL5);
\draw[<->] (nR4) to node[right] {\tiny$\tau_3$} (nR5);
\draw[<->] (nL5) to node[right] {\tiny$\tau_1$} (nL6);
\draw[<->] (nL5) to node[above] {\tiny$\tau_3$} (nR6);
\draw[<->] (nR5) to node[right] {\tiny$\tau_0$} (nR6);
\draw[<->] (nL6) to node[right] {\tiny$\tau_0$} (nL7);
\draw[<->] (nL6) to node[above] {\tiny$\tau_3$} (nR7);
\draw[<->] (nR6) to node[right] {\tiny$\tau_1$} (nR7);
\draw[<->] (nL7) to node[right] {\tiny$\tau_3$} (nL8);
\draw[<->] (nR7) to node[above] {\tiny$\tau_0$} (nL8);
\draw[<->] (nR7) to node[right] {\tiny$\tau_2$} (nR8);
\draw[<->] (nL8) to node[ right] {\tiny$\tau_2$} (n9);
\draw[<->] (nR8) to node[ right] {\tiny$\tau_0$} (n9);
\draw[<->] (n9) to node[right] {\tiny$\tau_1$} (n10);
\draw[<->] (n10) to node[right] {\tiny$\tau_0$} (n11);
}
$$
\end{frame}


%
%\only<4->{$
%\TikZ{[yscale=-1.5]
%\foreach \y in {4,5,6,7,8}{\coordinate (L\y) at (-1,\y*.55);	\coordinate (R\y) at (1,\y*.55);}
%\foreach \y in {1,2,3,9,10,11}{\coordinate (C\y) at (0,\y*.55);}
%\begin{scope}[every node/.style={inner sep = 1pt}]
%\node[] (n1) at (C1) {$\Row{1,2,3,4}$};
%\node (n2) at (C2) {$\Row{-1,2,3,4}$};
%\node (n3) at (C3) {$\Row{-2,1,3,4}$};
%\node (nL4) at (L4) {$\Row{-2,-1,3,4}$};
%\node (nR4) at (R4) {$\Row{-3,1,2,4}$};
%\node (nL5) at (L5) {$\Row{-3,-1,2,4}$};
%\node (nR5) at (R5) {$\Row{-4,1,2,3}$};
%\node (nL6) at (L6) {$\Row{-3,-2,1,4}$};
%\node (nR6) at (R6) {$\Row{-4,-1,2,3}$};
%\node (nL7) at (L7) {$\Row{-3,-2,-1,4}$};
%\node (nR7) at (R7) {$\Row{-4,-2,1,3}$};
%\node (nL8) at (L8) {$\Row{-4,-2,-1,3}$};
%\node (nR8) at (R8) {$\Row{-4,-3,1,2}$};
%\node (n9) at (C9) {$\Row{-4,-3,-1,2}$};
%\node (n10) at (C10) {$\Row{-4,-3,-2,1}$};
%\node (n11) at (C11) {$\Row{-4,-3,-2,-1}$};
%\end{scope}
%\draw[<->] (n1) to node[right] {\tiny$\tau_0$} (n2);
%\draw[<->] (n2) to node[right] {\tiny$\tau_1$} (n3);
%\draw[<->] (n3) to node[right] {\tiny$\tau_0$} (nL4);
%\draw[<->] (n3) to node[right] {\tiny$\tau_2$} (nR4);
%\draw[<->] (nL4) to node[right] {\tiny$\tau_2$} (nL5);
%\draw[<->] (nR4) to node[above] {\tiny$\tau_0$} (nL5);
%\draw[<->] (nR4) to node[right] {\tiny$\tau_3$} (nR5);
%\draw[<->] (nL5) to node[right] {\tiny$\tau_1$} (nL6);
%\draw[<->] (nL5) to node[above] {\tiny$\tau_3$} (nR6);
%\draw[<->] (nR5) to node[right] {\tiny$\tau_0$} (nR6);
%\draw[<->] (nL6) to node[right] {\tiny$\tau_0$} (nL7);
%\draw[<->] (nL6) to node[above] {\tiny$\tau_3$} (nR7);
%\draw[<->] (nR6) to node[right] {\tiny$\tau_1$} (nR7);
%\draw[<->] (nL7) to node[right] {\tiny$\tau_3$} (nL8);
%\draw[<->] (nR7) to node[above] {\tiny$\tau_0$} (nL8);
%\draw[<->] (nR7) to node[right] {\tiny$\tau_2$} (nR8);
%\draw[<->] (nL8) to node[ right] {\tiny$\tau_2$} (n9);
%\draw[<->] (nR8) to node[ right] {\tiny$\tau_0$} (n9);
%\draw[<->] (n9) to node[right] {\tiny$\tau_1$} (n10);
%\draw[<->] (n10) to node[right] {\tiny$\tau_0$} (n11);
%}$}

%
%
%
%\begin{frame} \end{frame}
%\begin{frame} \end{frame}
%
%
%\begin{frame}
%
%
%\begin{tikzpicture}[xscale=-.75, yscale=-.75]
%\draw [dashed] (5,2) node[above left, inner sep=1pt] {$a$} -- (2,8) node[below right, inner sep=1pt] {$a$};
%\draw [dashed] (3.5,-1)	node[above left, inner sep=1pt] {$b$} -- (-1,8) node[below right, inner sep=1pt] {$b$};
%\draw [dashed] (-.5,-1)	node[above right, inner sep=1pt] {$k$} -- (4,8) node[below left, inner sep=1pt] {$k$};
%\foreach\k [evaluate=\k as \w using \k*.5+4.5,remember=\k as \lasty, count=\c from 3] in {0,1,...,7}{
%	\foreach \x [remember=\x as \lastx] in {0,...,\w}{
%	% Vertices: 
%	\ifnum \x<\c \else \breakforeach \fi
%	\ifnum \x<4
%		\ifnum \x<\k \TLNode{SumToL}
%			\else  \TLNode{FULL}
%		\fi
%	\else
%		\ifnum \x > 6 
%			\TLNode{Overlapping} 
%		\else
%			\ifnum \x < \k  \TLNode{SumToA} \else 
%				\ifnum \x> \c  \else	\TLNode{SumToBKL}	\fi
%			\fi
%		\fi
%	\fi
%	% Edges:
%	\ifnum \k>0
%		\ifnum \x > 0
%			\draw (\x\k) to (\lastx\lasty);
%			\fi
%		\ifnum \x<\c 
%			\ifnum \x < \w
%				\draw (\x\k) to (\x\lasty);
%			\fi
%		\fi
%	\fi
%	}}
%	\draw (64)--(63) ;
%\node[FULL] (a) at (1.5,-1.5){\scriptsize$0$};
%\foreach \x in {0,1,...,3}{\draw (a)--(\x0);}
%\foreach \k in {0,1,...,7}{\node[left] at (5.5,\k){\scriptsize$k=\k$}; }
%\node[SumToL, label=right:{ $= \TikZ{[xscale=.2, yscale=-.2] 
%				\Part{7,4} 
%				\draw[red, |-|] (0,-1) to node[midway, above]{\tiny $k+a+b-\ell$} (7,-1);
%				\draw[red, |-|] (0,3) to node[midway, below]{\tiny $\ell$} (4,3);	
%			}$}] at (-4.25,2) {\scriptsize$\ell$};
%\end{tikzpicture}
%
%\end{frame}
%

\begin{frame}~\end{frame}
\end{document}