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%% 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\cPole[#1][#2,#3][#4]{%1=horizontal location, 2 = bottom, 3=top
	\filldraw[white] (#1+.175,#2+.05) rectangle (#1-.175,#3-.05);
	\shade[left color=#4,right color=white] (#1+.1,#2) rectangle (#1-.1,#3);
	\draw[] (#1+.1,#2) to (#1+.1,#3) (#1-.1,#2) to (#1-.1,#3) ;}


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	\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 locations, 2 = bottom, 3=top, 4=k 
	\Tops[#1][#3][#4]
	\Bottoms[#1][#2][#4]
	}

	
\def\cTops[#1][#2][#3]{%1=pole locations, 2=top, 3=k 
	\foreach\x/\y in {#1}{
		\shade[left color=\y,right color=white] (\x+.1,#2) rectangle (\x-.1,#2+.15);
		\draw (\x+.1,#2) -- (\x+.1,#2+.15) (\x-.1,#2) -- (\x-.1,#2+.15) ;
		\draw[fill=white] (\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\cBottoms[#1][#2][#3]{%1=pole locations, 2 = bottom, 3=top 
	\foreach\x/\y in {#1}{
		\shade[left color=\y,right color=white] (\x+.1,#2) rectangle (\x-.1,#2-.1);
		\draw (\x+.1,#2) -- (\x+.1,#2-.1) (\x-.1,#2) -- (\x-.1,#2-.1) ;
		\shade[left color=\y,right color=white] (\x+.1,#2-.1) arc (0:-180:1mm and .5mm);
		\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\cCaps[#1][#2,#3][#4]{%1=pole location, 2 = bottom, 3=top, 4=k 
	\cTops[#1][#3][#4]
	\cBottoms[#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) ;
}
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\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};		}

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		\fi}
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   	\foreach \y in {0, ..., \x} {\draw (\y,\value{r})--(\y,\value{r}-1);}
	\addtocounter{r}{1}
 }}
 
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\TikZ{[xscale=\PartUNIT, yscale=-\PartUNIT] 
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}
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\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);
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\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);
}
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\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);
}
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\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);
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	 \foreach \x [count = \c from 0] in {#1}{
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}
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\begin{tikzpicture}[scale=\RowUNIT]
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}
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}

	\filldraw [red] (#1-1,1) circle (4pt);
 \end{tikzpicture}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Brattelli diagram
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\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}
}

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	\foreach \x in {1, ..., #3}{
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		\node[V] (t\x) at (\x*.75+.25, #2) {};}}
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	\foreach \x in {1, ..., #3}{
		\node[V] (b\x) at (\x*.75+.25, #1){};	
		\node[V] (t\x) at (\x*.75+.25, #2) {};}}
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		\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{Combinatorics and representation theory of Temperley-Lieb algebras}
\author[Zajj Daugherty]{Zajj Daugherty\\~\\{\small The City College of New York}\\{\small \& CUNY Graduate Center}}
\date{October 1, 2019}





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



%%%%%%%%%%%%
%%%%%%%%%%%%
\begin{frame}{Motivating example: Schur-Weyl Duality}
The {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.
$$\RED{g} \cdot (v_1 \otimes v_2 \otimes \cdots \otimes v_k) = \RED{g} v_1 \otimes \RED{g} v_2 \otimes \cdots \otimes \RED{g} v_k.$$
\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!  
$$\begin{minipage}{4.5cm}\begin{tikzpicture}[line width=.75pt, scale=.9]
\foreach \x in {1,...,5}{
\node[bV](t\x) at (\x,1){}; 
\node[bV, label=below:{\small \color{red}$g$\alert{$v_{\x}$}}](b\x) at (\x,0){}; 
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \color{themecolor} $\otimes$};
\node at (\x+.5, -.3) {\small \color{themecolor} $\otimes$};
}
\node[above] at (t1) {\small \color{red}$g$\alert{$v_{2}$}};
\node[above] at (t2) {\small \color{red}$g$\alert{$v_{4}$}};
\node[above] at (t3) {\small \color{red}$g$\alert{$v_{1}$}};
\node[above] at (t4) {\small \color{red}$g$\alert{$v_{5}$}};
\node[above] at (t5) {\small \color{red}$g$\alert{$v_{3}$}};
\draw[->] (b2)--(t1); 
\draw[->] (b4)--(t2);
\draw[->] (b1)--(t3); 
\draw[->] (b5)--(t4); 
\draw[->] (b3)--(t5); 
\end{tikzpicture}\end{minipage} \text{ vs. } 
\begin{minipage}{4.5cm}\begin{tikzpicture}[line width=.75pt, scale=.9]
\foreach \x in {1,...,5}{
\node[bV](t\x) at (\x,1){}; 
\node[bV, label=below:{\small \alert{$v_{\x}$}}](b\x) at (\x,0){}; 
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \alert{$\otimes$}};
\node at (\x+.5, -.3) {\small \alert{$\otimes$}};
}
\node[above] at (t1) {\small \color{red}$g$\alert{$v_{2}$}};
\node[above] at (t2) {\small \color{red}$g$\alert{$v_{4}$}};
\node[above] at (t3) {\small \color{red}$g$\alert{$v_{1}$}};
\node[above] at (t4) {\small \color{red}$g$\alert{$v_{5}$}};
\node[above] at (t5) {\small \color{red}$g$\alert{$v_{3}$}};
\draw[->] (b2)--(t1); 
\draw[->] (b4)--(t2);
\draw[->] (b1)--(t3); 
\draw[->] (b5)--(t4); 
\draw[->] (b3)--(t5); \end{tikzpicture}\end{minipage}$$

\end{enumerate}
\end{frame}

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



\begin{frame}{{Motivating example: Schur-Weyl Duality}} 
${}\!$
{\small Consider the representations induced  by these commuting actions,
\\ \smallskip
\centerline{$\displaystyle
\pi: \CC S_k \to \End(\left( \CC^n\right)^{\otimes k}) \quad \text{and} \quad
\rho: \CC \GL_n \to \End(\left( \CC^n\right)^{\otimes k}) .
$} \smallskip
}

\smallskip

\alert{Thm. (Schur 1901)}
$$\!\!\!\underbrace{\End_{\GL_n}\left(\left( \CC^n\right)^{\otimes k} \right)}_{\text{\scriptsize (all linear maps that} \atop \text{\scriptsize commute with $\GL_n$)}} 
	= \underbrace{\pi(\CC S_k)\phantom{\Big|}\!\!}_{\text{\scriptsize (img of $S_k$} \atop\text{\scriptsize action)}}
	  \text{ and }  \End_{S_k}\left( \left( \CC^n\right)^{\otimes k}\right) 
	 = \underbrace{\rho(\CC \GL_n)\phantom{\Big|}\!\!}_{\text{\scriptsize (img of $\GL_n$} \atop\text{\scriptsize action)}}
\!\!. $$%
\uncover<2->{\begin{block}{Powerful consequence: a duality between representations}
		The double-centralizer relationship produces
			$$\left( \CC^n\right)^{\otimes k}  \cong \bigoplus_{\tiny \lambda \vdash k \atop \ell(\lambda) \leq n }  G^\lambda \otimes S^\lambda
			\quad \text{ as a $\GL_n$-$S_k$ bimodule,}$$
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}{Temperley-Lieb algebras}
Caution! The representation \\ \smallskip
\centerline{$\displaystyle
\pi: \CC S_k \to \End \left(\left(\CC^n\right)^{\otimes k}\right) 
$} \smallskip
is not always injective! \hfill \fbox{\alert{Thm.} $\ker(\pi) \neq 0$ when $n<k$.} \pause 

\alert{Case $n = 2$:} Define
\\ \smallskip
\centerline{$\displaystyle
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	}
~=~
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw (t1) to (b1) (t2) to (b2);
	}
~-~
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw (t1) to (b2) (t2) to (b1);
	}
$.} \smallskip\pause 
Then in $\CC S_k$ (for general $k$),
\begin{align*}
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (m2) to (m1) (m1) to (m2) (b1) to (b2);
	}
= \left(\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	}\right)^2
\uncover<4->{=\left(\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[red] (t1) to (b1) (t2) to (b2);
	}
-
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[blue] (t1) to (b2) (t2) to (b1);
	}\right)^2}
&\uncover<5->{=
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[red] (t1) to (m1) (t2) to (m2);
	\draw[red] (m1) to (b1) (m2) to (b2);
	}
-
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[red] (t1) to (m1) (t2) to (m2);
	\draw[blue] (m1) to (b2) (m2) to (b1);
	}
-
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[blue] (t1) to (m2) (t2) to (m1);
	\draw[red] (m1) to (b1) (m2) to (b2);
	}
+
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[blue] (t1) to (m2) to (b1) (t2) to (m1) to (b2);
	}}\\
&\uncover<6->{=2\left(\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[red] (t1) to (b1) (t2) to (b2);
	}
-
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[blue] (t1) to (b2) (t2) to (b1);
	}\right)}
\uncover<7->{= 2\left(\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	}\right).}
\end{align*}
\uncover<7->{{\small \emph{Because $\TikZ{[xscale=.25, yscale=.35]
	\foreach\x in {1,2}{
		\node[V] (t\x) at (\x,1){};
		\node[V] (b\x) at (\x,0){};}
	\draw [bend left=40] (t2) to (t1) (b1) to (b2);
	}$
	is (2$\times$) the projection onto the sign representation for $S_2$.}}}


\end{frame}



%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Temperley-Lieb algebras}%${}\!\!$
%\footnotesize
Caution! The representation \\ 
\centerline{$\displaystyle
\pi: \CC S_k \to \End \left(\left(\CC^n\right)^{\otimes k}\right) 
$} 
is not always injective! \hfill \fbox{\alert{Thm.} $\ker(\pi) \neq 0$ when $n<k$.} 

\bigskip

\alert{Case $n = 2$:} Define
\\ \smallskip
\centerline{$\displaystyle
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	}
~=~
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw (t1) to (b1) (t2) to (b2);
	}
~-~
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw (t1) to (b2) (t2) to (b1);
	}
$.} \smallskip


\emph{Only} true for  $n\le 2$:  
\\ \medskip
\centerline{$\displaystyle
\TikZ{[Diagram]
	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
		\node[bV] (t\x) at (\x,3){};
		\node[bV] (M\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60]
		 (t2) to (t1) (M1) to (M2)
		 (M3) to (M2) (m2) to (m3)
		 (m2) to (m1) (b1) to (b2);
	\draw (t3) to (M3) (M1) to (m1) (m3) to (b3);
	}
	~-~
	\TikZ{[Diagram]
	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	\draw (t3) to (b3);
	} ~\in \ker(\pi)
	\qquad 
\TikZ{[Diagram]
	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
		\node[bV] (t\x) at (\x,3){};
		\node[bV] (M\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60]
		 (t3) to (t2) (M2) to (M3)
		 (M2) to (M1) (m1) to (m2)
		 (m3) to (m2) (b2) to (b3);
	\draw (t1) to (M1) (M3) to (m3) (m1) to (b1);
	}
	~-~
	\TikZ{[Diagram]
	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t3) to (t2) (b2) to (b3);
	\draw (t1) to (b1);
	} ~\in \ker(\pi)	
$} \smallskip
%\\ \smallskip
%\centerline{$\displaystyle
%\TikZ{[Diagram]
%	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
%		\node[bV, label={above,rotate=30}:{\tiny$\y$}] (t\x) at (\x,3){};
%		\node[bV] (M\x) at (\x,2){};
%		\node[bV] (m\x) at (\x,1){};
%		\node[bV, label={below,rotate=30}:{\tiny$\y$}] (b\x) at (\x,0){};}
%	\draw [bend left=60]
%		 (t2) to (t1) (M1) to (M2)
%		 (M3) to (M2) (m2) to (m3)
%		 (m2) to (m1) (b1) to (b2);
%	\draw (t3) to (M3) (M1) to (m1) (m3) to (b3);
%	}\!\!\!\!
%	-\!\!
%	\TikZ{[Diagram]
%	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
%		\node[bV, label={above,rotate=30}:{\tiny$\y$}] (t\x) at (\x,1){};
%		\node[bV, label={below,rotate=30}:{\tiny$\y$}] (b\x) at (\x,0){};}
%	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
%	\draw (t3) to (b3);
%	} \!\!\in \ker(\pi)
%	\quad 
%\TikZ{[Diagram]
%	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
%		\node[bV, label={above,rotate=30}:{\tiny$\y$}] (t\x) at (\x,3){};
%		\node[bV] (M\x) at (\x,2){};
%		\node[bV] (m\x) at (\x,1){};
%		\node[bV, label={below,rotate=30}:{\tiny$\y$}] (b\x) at (\x,0){};}
%	\draw [bend left=60]
%		 (t3) to (t2) (M2) to (M3)
%		 (M2) to (M1) (m1) to (m2)
%		 (m3) to (m2) (b2) to (b3);
%	\draw (t1) to (M1) (M3) to (m3) (m1) to (b1);
%	}\!\!\!\!
%	-\!\!
%	\TikZ{[Diagram]
%	\foreach\x/\y in {1/i,2/i\!+\!1,3/i\!+\!2}{
%		\node[bV, label={above,rotate=30}:{\tiny$\y$}] (t\x) at (\x,1){};
%		\node[bV, label={below,rotate=30}:{\tiny$\y$}] (b\x) at (\x,0){};}
%	\draw [bend left=60] (t3) to (t2) (b2) to (b3);
%	\draw (t1) to (b1);
%	} \!\!\in \ker(\pi)	
%$} \smallskip



~\bigskip


\end{frame}



\begin{frame}{Temperley-Lieb algebras}
Fix $\delta \in \CC$. The \emph{Temperley-Lieb algebra $TL_k$} is a diagram algebra generated over $\CC$ by diagrams \uncover<2->{\hfill \fbox{\small \alert{Basis:} all non-crossing diagrams}}
\smallskip
\centerline{$\displaystyle
e_i = 
\TikZ{[Diagram]
	\foreach\x in {1,3,4,5,6,8}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\node[above] at (t4) {\tiny$i$}; \node[below] at (b4) {\tiny$i$}; 
	\foreach \x in {1,3,6,8}{\draw (b\x) to (t\x);}
	\draw[bend right=70] (t4) to (t5) (b5) to (b4); 
	\node at (2,.5) {\small $\cdots$}; \node at (7,.5) {\small $\cdots$};
	},
	\qquad 
$ for $i=1, \dots, k-1$,} \smallskip
\uncover<1->{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<1->{\begin{matrix}e_i e_{i \pm 1} e_i = e_i \\ \text{\scriptsize for  $1 \leq i \leq k-1$,} \end{matrix}}
&
\uncover<1->{
\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 (1,0) to (1,1) (3,1) to (3,2) (1,2) to (1,3);
	\draw[bend left=50]  (2,0) to (3,0) (3,1) to (2,1) (1,1) to (2,1) (2,2) to (1,2) (2,2) to (3,2) (3,3) to (2,3);
	}
	=
	\TikZ{[diagram]
	\draw (1,0) to (1,1.5);
	\draw[bend left=80]  (2,0) to (3,0) (3,1.5) to (2,1.5);
	}$\ }
	\qquad \uncover<3->{(\star)}
}
\\~\\
\uncover<1->{e_i^2 = \delta_i e_i. }
& 
\uncover<1->{\fbox{$\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);
}$}}

\end{array}$$

\uncover<3->{
\!\!\alert{Thm.} The quotient of $\CC S_k$ by relations ($\star$) factors through the representation 
\\ 
\centerline{$\displaystyle
\pi: \CC S_k \to \End\left( \left(\CC^2\right)^{\otimes k} \right)
$}  (i.e.\ when $\delta = 2$, $TL_k$ centralizes the action of $\GL_2$ on $\left(\CC^2\right)^{\otimes k}$). 

}


\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 $\fg$ (deform the Lie algebra by a parameter $q$). \pause

$\cU\otimes \cU$ has an invertible element $\cR$ called an \alert{R-matrix} 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  \only<1-3>{\alert{$\cB_k$}}\only<4>{\alert{$\cB_k^{(1)}$}}\only<5->{\alert{$\cB_k^{(2)}$}} 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}\small
The \alert{type-A Hecke algebra} is the quotient of the group algebra of the \alert{braid group $\cB_k$} by relations 
\begin{equation}\tag{$*$}\label{1}
\TikZ{[Diagram, scale=.75]
	\Cross[1,2][2,1];
	\Cross[1,1][2,0];
\foreach \x in {1,2}{
	\node[bV]  at (\x, 2) {};
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}
= (q-q^{-1})\TikZ{[Diagram, scale=.75]
	\Cross[1,1][2,0];
\foreach \x in {1,2}{
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}
	~+~
	\TikZ{[Diagram, scale=.75]
	\draw (1,0) to (1,1) (2,0) to (2,1);
\foreach \x in {1,2}{
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}.\end{equation}

\only<1-3>{\alert{Thm.} The action of $\CC B_k$ on $V^{\otimes k}$ factors through the quotient by \eqref{1} when $V = \CC^n$ and $\fg = \fgl_n$ or $\fsl_n$.}


\pause
\medskip

\alert{\hrule}

\medskip

The \alert{affine type-$GL_k$ Hecke algebra} is the quotient of the group algebra of the \alert{one-pole braid group $\cB_k^{(1)}$} by relations \eqref{1}.

\only<1-3>{\smallskip

\alert{Thm.} The action of $\CC B_k^{(1)}$ on $M \otimes V^{\otimes k}$ factors through the quotient by \eqref{1} when $V = \CC^n$ and $\fg = \fgl_n$ or $\fsl_n$.}


\pause
\medskip

\alert{\hrule}

\medskip

The \alert{affine type-C Hecke algebra} is the quotient of the group algebra of the \alert{two-pole braid group $\cB_k^{(2)}$} by relations \eqref{1}, 
\begin{equation}\tag{$**$}\label{2}
\TikZ{[Diagram, yscale=.6]
	\Under[1,1][.2,.5]
	\Pole[.5][0.2,1] 
	\Over[1,.1][.2,.5]
	\Under[1,1.9][.2,1.5]
	\Pole[.5][1,1.8] 
	\Over[1,1][.2,1.5]
	\Caps[.5][0.2,1.8][1]
	}
=a\TikZ{[Diagram]
	\Under[1,.9][.2,.5]
	\Pole[.5][0.2,0.8] 
	\Over[1,.1][.2,.5]
	\Caps[.5][0.2,0.8][1]
	} +  \TikZ{[Diagram]
	\draw(1,.2) to (1,.8);
	\Pole[.5][0.2,0.8] 
	\Caps[.5][0.2,0.8][1]
	}
\quad\text{and}\quad
\TikZ{[Diagram, yscale=.6, xscale=-1]
	\Under[1,1][.2,.5]
	\Pole[.5][0.2,1] 
	\Over[1,.1][.2,.5]
	\Under[1,1.9][.2,1.5]
	\Pole[.5][1,1.8] 
	\Over[1,1][.2,1.5]
	\Caps[.5][0.2,1.8][1]
	}
=b\TikZ{[Diagram, xscale=-1]
	\Under[1,.9][.2,.5]
	\Pole[.5][0.2,0.8] 
	\Over[1,.1][.2,.5]
	\Caps[.5][0.2,0.8][1]
	} +  \TikZ{[Diagram, xscale=-1]
	\draw(1,.2) to (1,.8);
	\Pole[.5][0.2,0.8] 
	\Caps[.5][0.2,0.8][1]
	}
\end{equation}

\only<1-3>{\alert{Thm.} The action of $\CC B_k^{(2)}$ on $M \otimes V^{\otimes k} \otimes N$ factors through the quotient by \eqref{1} and \eqref{2} when $V = \CC^n$, $M$ and $N$ are ``rectangular'', and $\fg = \fgl_n$ or $\fsl_n$.}



\pause

\medskip

\alert{\hrule}

\medskip
\alert{``Type what-now?''}\\
Dynkin diagrams:
\\ \smallskip
\centerline{$\displaystyle
\TikZ{[scale=.7]
\foreach \x in {1,2,4}{\node[draw, circle, inner sep=2pt] (c\x) at (\x,0) {};}
\draw (c1) to (c2) (c2) to (2.5,0) (c4) to (3.5,0);
\node at (3,0) {$\cdots$};
\node at (2.5,.75) {Type A};
}
\qquad
\TikZ{[scale=.7]
\foreach \x in {0,1,2,4}{\node[draw, circle, inner sep=2pt] (c\x) at (\x,0) {};}
\draw (c1) to (c2) (c2) to (2.5,0) (c4) to (3.5,0);
\draw[double distance=2pt](c0) to (c1);
\node at (3,0) {$\cdots$};
\node at (2,.75) {Affine Type $GL$};
}
\qquad
\alert<5->{\TikZ{[scale=.7]
\foreach \x in {0,1,2,4,5}{\node[draw, circle, inner sep=2pt] (c\x) at (\x,0) {};}
\draw (c1) to (c2) (c2) to (2.5,0) (c4) to (3.5,0);
\draw[double distance=2pt](c0) to (c1) (c4) to (c5);
\node at (3,0) {$\cdots$};
\node at (2.5,.75) {Affine Type C};
}}
$} \smallskip

\end{frame}


\begin{frame}{}
The \alert{two-pole/affine type-C  braid group} is the group $\cB_k^{(2)}$ generated by 
$ T_0,  T_1,\ldots,  T_k$,
with relations 
\begin{equation*}\label{Bdefrels}
\begin{tikzpicture}
	\foreach \x in {0, 1, 2, 4,5,6}{
		\node[draw, circle, inner sep=2pt] (\x) at (\x,0) {};
		}
	\foreach \x in {0, 1, 2}
		{\node[label=above:$ T_\x$] at (\x){};}
	\node[label=above:$ T_{k-2}$] at (4){};
	\node[label=above:$ T_{k-1}$] at (5){};
	\node[label=above:$ T_{k}$] at (6){};
	\draw[double distance = 2pt] (0)--(1) (5)--(6);
	\draw (1)--(2)--(2.5,0) (3.5,0)--(4)--(5) ;
%	\draw[dashed] (2)--(4);
	\node at (3,0) {$\cdots$};
	\only<3>{
		\node[draw, circle, inner sep=2pt, structure, line width = 1pt, ] at (1) {};
		\node[draw, circle, inner sep=2pt, structure, line width = 1pt, ] at (2) {};
		\draw[structure, line width = 1pt, ] (1)--(2);
	}
	\only<4>{
		\node[draw, line width = 1pt, circle, inner sep=2pt, structure] at (1) {};
		\node[draw, line width = 1pt, circle, inner sep=2pt, structure] at (0) {};
		\draw[structure, double distance = 2pt, line width = 1pt] (1)--(0);
	}
\end{tikzpicture}\end{equation*}


\uncover<2->{
Pictorially, \only<1-2>{the generators of $\cB_k^{(2)}$ are identified with the diagrams
$$
{\def\TOP{2}\def\K{6}
\begin{tikzpicture}[Diagram, yscale=.5]
\node at (0,.5*\TOP) {$ T_k=$};
%%%%%%%%%%%%%
\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%
\Pole[.4][0,2]
\Under[\K,0][\K+1,1]
\Pole[\K+.5][0,1]
\Pole[\K+.5][1,2]
\Over[\K+1,1][\K,2]
 \foreach \x in {1,...,5} {
	 \draw (\x,0) -- (\x,\TOP);
	 }
\Caps[.4,\K+.5][0,2][\K]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{1.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {,};
\end{tikzpicture}
	\qquad 
\begin{tikzpicture}[Diagram, yscale=.5]
\node at (0,.5*\TOP) {$ T_0=$};
%%%%%%%%%%%%%
\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%
	\Pole[\K+.5][0,2]
	\Pole[.4][0,1]
	\Over[1,0][0,1]
	\Under[0,1][1,2]
	\Pole[.4][1,2]
	 \foreach \x in {2,...,\K} {
	 \draw (\x,0) -- (\x,\TOP);
	 }
	\Caps[.4,\K+.5][0,2][\K]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{1.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {,};
\end{tikzpicture}
}$$
and
$$
{\def\TOP{2} \def\K{6}
\begin{tikzpicture}[Diagram, yscale=.5]
\node at (0,.5*\TOP) {$ T_i=$};
%%%%%%%%%%%%%
\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%
	\Pole[\K+.5][0,2]
	\Pole[.4][0,2]
	\Under[3,0][4,2]
	\Over[4,0][3,2]
	 \foreach \x in {1,2,5,\K} {
		 \draw (\x,0) -- (\x,\TOP);
		 }
	\Caps[.4,\K+.5][0,2][\K]
	\Label[0-.2,\TOP+.25][3][\footnotesize $i$]
	\Label[0-.2,\TOP+.2][4][\footnotesize$i$+1]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{1.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {\qquad for $i=1, \dots, k-1$.};
\end{tikzpicture}
}
$$
}}
\only<3->{
\uncover<3->{
$$
{\def\TOP{2.5}\begin{tikzpicture}[line width=1pt , scale=.7]
	\node[left] at (.5,.75) {$T_i T_{i+1} T_i ~=$};
	\Under[2,1.5][1,.75]
	\Under[3,1.2][1,0]
	\Over[1,1.5][3,.3]
	\Over[1,.75][2,0]
	\draw(3,1.5)--(3,1.2) (3,.3)--(3,0);
	\Nodes[0][3]
	\Nodes[1.5][3]
\pgftransformxshift{3cm};
\node at (1,.75) {$=$};
\pgftransformxshift{1cm};
	\Under[3,.75][2,0]
	\Under[3,1.5][1,.3]
	\Over[1,1.2][3,0]
	\Over[2,1.5][3,.75]
	\draw(1,1.5)--(1,1.2) (1,.3)--(1,0);
	\Nodes[0][3]
	\Nodes[1.5][3]
\pgftransformxshift{3.5cm};
	\node[right] at (0,.75) {$= ~T_{i+1} T_i T_{i+1}$};
\end{tikzpicture}}$$
}%
\uncover<4->{
$${\def\TOP{2.5}\begin{tikzpicture}[Diagram, yscale=.75]
\node[left] at (-.5,.5*\TOP) {$T_1 T_0 T_1 T_0~=$};
	\Under[.2,.5][2,1.5]
	\Under[0,1.5][2,2.5]
	\Pole[.5][0,\TOP]
	\Over[0,1.5][2,.7]
	\Over[1,0][.2,.5]
	\draw (2,.7)--(2,0);
	\Over[1,2.5][2,1.7]
	\draw (2,1.7)--(2,1.5);
	\Caps[.5][0,\TOP][2]
\pgftransformxshift{3cm};
\node at (0,.5*\TOP) {$=$};
\pgftransformxshift{1cm};
	\Under[1,2.5][.2,2]
	\Under[2,2][0,1]
	\Under[2,.7][1,0]
	\Pole[.5][0,\TOP]
	\Over[.2,2][2,1.2]
	\Over[0,1][2,0]
	\draw (2,2.5)--(2,2);
%	\Over[1,2][2,1.2]
	\draw (2,1.2)--(2,.7);
	\Caps[.5][0,\TOP][2]
\pgftransformxshift{2.5cm};
\node[right] at (0,.5*\TOP) {$= ~ T_0 T_1 T_0 T_1$};
\end{tikzpicture}}$$

\centerline{\small (similar picture for $T_k T_{k-1} T_k T_{k-1} = T_{k-1} T_k T_{k-1} T_k$)}
}}

\end{frame}




%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Back to Temperley-Lieb algebras}
\small\only<6-7>{\scriptsize}

The \alert{type-A Hecke algebra} $HA_k$ is the quotient of the group algebra of the \alert{braid group $\cB_k$} by relations 
\begin{equation}\tag{$*$}
\TikZ{[Diagram, xscale=.75, yscale=.5]
	\Cross[1,2][2,1];
	\Cross[1,1][2,0];
\foreach \x in {1,2}{
	\node[bV]  at (\x, 2) {};
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}
= (q-q^{-1})\TikZ{[Diagram, scale=.75]
	\Cross[1,1][2,0];
\foreach \x in {1,2}{
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}
	~+~
	\TikZ{[Diagram, scale=.75]
	\draw (1,0) to (1,1) (2,0) to (2,1);
\foreach \x in {1,2}{
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}.\end{equation}

\alert{Thm.} The action of $\CC B_k$ on $V^{\otimes k}$ factors through the quotient by \eqref{1} when $V = \CC^n$ and $\fg = \fgl_n$ or $\fsl_n$.

\pause\small
\only<1-7>{\medskip

\alert{\hrule}

\medskip}\only<8->{\bigskip

\alert{\hrule}

\bigskip}

\alert{Case $n = 2$:} Define
\begin{equation}\tag{$\diamond$}\label{3}
\TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	}
~=~
q\ \TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw (t1) to (b1) (t2) to (b2);
	}
~-~
 \TikZ{[Diagram]
	\Cross[1,1][2,0]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	}.
\end{equation}
\pause
\only<1-7>{Then in $HA_k$ (for general $k$),
\scriptsize
\begin{align*}
\TikZ{[Diagram, scale=.9]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (m2) to (m1) (m1) to (m2) (b1) to (b2);
	}
&= \left(\TikZ{[Diagram, scale=.9]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	}\right)^2
\uncover<4->{=\left(
	q\ \TikZ{[Diagram, scale=.9]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw (t1) to (b1) (t2) to (b2);
	}
-
\TikZ{[Diagram, scale=.9]
	\Cross[1,1][2,0]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	}\right)^2}
\uncover<5->{=
q^2\ \TikZ{[Diagram, scale=.9]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[] (t1) to (m1) (t2) to (m2);
	\draw[] (m1) to (b1) (m2) to (b2);
	}
-
q\ \TikZ{[Diagram, scale=.9]
	\Cross[1,1][2,0]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[] (t1) to (m1) (t2) to (m2);
	}
-
q\ \TikZ{[Diagram, scale=.9]
	\Cross[1,2][2,1]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw[] (m1) to (b1) (m2) to (b2);
	}
+
 \TikZ{[Diagram, scale=.9]
	\Cross[1,1][2,0]
	\Cross[1,2][2,1]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,2){};
		\node[bV] (m\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	}}\\
&\uncover<6->{=
	q^2\ \TikZ{[Diagram]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw (t1) to (b1) (t2) to (b2);
	}
	-q\ \TikZ{[Diagram]
	\Cross[1,1][2,0]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	}
	+ \left( 
	(q-q^{-1})\TikZ{[Diagram, scale=.75]
	\Cross[1,1][2,0];
\foreach \x in {1,2}{
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}
	~+~
	\TikZ{[Diagram, scale=.75]
	\draw (1,0) to (1,1) (2,0) to (2,1);
\foreach \x in {1,2}{
	\node[bV]  at (\x, 1) {};
	\node[bV]  at (\x, 0) {};}}
	\right)
	}
\uncover<7->{=(q+q^{-1})\TikZ{[Diagram, scale=.9]
	\foreach\x in {1,2}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left=60] (t2) to (t1) (b1) to (b2);
	}.}
\end{align*}
\uncover<7->{{\small \emph{Because $\TikZ{[xscale=.25, yscale=.35]
	\foreach\x in {1,2}{
		\node[V] (t\x) at (\x,1){};
		\node[V] (b\x) at (\x,0){};}
	\draw [bend left=40] (t2) to (t1) (b1) to (b2);
	}$
	is $(q+q^{-1})\times$(proj.\ onto  sign representation for $HA_2$).}}}}
\only<8->{

\bigskip

\alert{Thm.} Using the identification in \eqref{3}, the action of $HA_k$ on $(\CC^2)^{\otimes k}$ factors through the Temperley-Lieb quotient when $\delta = q+q^{-1} = [2]_q$, i.e.\ $TL_k$ centralizes $\cU_q \fgl_2$ and $\cU_q \fsl_2$ in $\End((\CC^2)^{\otimes k})$ when $\TikZ{[line width=.75pt] \draw (0,0) circle (0.15cm);} = [2]_q$. 


\bigskip
}

\end{frame}




\begin{frame}

\footnotesize
\def\TOP{1.5}
\def\SCALE{.4}

\centerline{$\displaystyle
\TikZ{[diagram]
	\draw[densely dotted, alert] (.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); 
	}
	= q \ \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[1,1][1.75,0]
			\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){};	
			}
			\qquad\qquad
\TikZ{[diagram]
	\draw[line width=1.5pt, alert] (.5,0) to (.5,1.25);
	\draw[densely dotted, alert] (.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); 
	}
	=     q_0\ \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]
		}
		\qquad\qquad
\TikZ{[diagram]
	\draw[line width=1.5pt, alert] (1.5,0) to (1.5,1.25);
	\draw[densely dotted, alert] (.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); 
	}
	=     q_k\ \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]
			}
$} \smallskip


\def\SCALE{1}
\centerline{
\begin{tabular}{|c|c|c|c|}\hline
tensor & centralizer& centralizer & centralizer\\
space & of $\cU_q \fg$ & of $\cU_q \fgl_n$ &  of $\cU_q \fgl_2$\\ \hline\hline
\phantom{\Big|}$V^{\otimes k}\phantom{\Big|}$  
	& Braids on $k$ strands 
	& Type-A Hecke 
	& Temperley-Lieb \\
& $\TikZ{[line width=.75pt, scale=.5]
	\draw (2,1.5).. controls (4,1) .. (5,0);
	\Under[1,1.5][1.75,.75];
	\Under[4,1.5][4.5,.75];
		\Under[1.25,.75][2,0];
		\Over[3,1.5][1.25,.75];
	\Over[5,1.5][4,.75];
	\Over[1.75,.75][1,0];
	\Over[4,.75][4,0];
	\Over[4.5,.75][3,0];
	\foreach \x in {1,...,5}{
		\draw (\x,0)  to (\x,-.05) node[V]{};
		\draw (\x,\TOP)  to (\x,\TOP+.05) node[V]{};
	}}$ 
	& (twist relations) 
	& $\TikZ{[diagram, scale=.5]
		\NFrame[0,1.5][5]
		\draw[bend right=50] 
			 (b5) to (b4) 
			(t2) to (t3);
		\draw (t1) to (b1) (t4) to (b2) (t5) to (b3); 
		}$
	\\\hline\hline
%%%%%%%%%%%%%%%%%%%%%%%%
$\phantom{\Big|}M \otimes V^{\otimes k}\phantom{\Big|}$ 
	& One-pole braids & Affine type-$\GL$ Hecke & 1-boundary TL\\
	& $\TikZ{[line width=.75pt, scale=.5]
		\draw (2,1.5).. controls (4,1) .. (5,0);
		\Under[1,1.5][1.75,.75];
		\Under[4,1.5][4.5,.75];
		\Under[.2,.75][2,0];
		\Pole[.45][0,\TOP]
		\Over[3,1.5][.2,.75];
		\Over[5,1.5][4,.75];
		\Over[1.75,.75][1,0];
		\Over[4,.75][4,0];
		\Over[4.5,.75][3,0];
		\Caps[.45][0,\TOP][5]}$ 
		&(twist relations) 
		&  $\TikZ{[diagram, scale=.5]
		\LFrame[0,1.5][5]
		\node[wV] (L1) at (.5,.5) {}; 
		\node[wV] (L2) at (.5,1) {}; 
		\draw[bend right=50] 
			(L2) to (t1) (b1) to (L1) (b5) to (b4) 
			(t2) to (t3);
		\draw (t4) to (b2) (t5) to (b3); 
		}$
		\\\hline\hline
%%%%%%%%%%%%%%%%%%%%%%%%
$\!\!\phantom{\Big|}M \otimes V^{\otimes k} \otimes N\phantom{\Big|}\!\!$ 
	& Two-pole braids  & Affine type-C Hecke & 2-boundary TL\\
	& $\TikZ{[line width=.75pt, scale=.5]
		\draw (2,1.5).. controls (4,1) .. (5,0);
		\Under[1,1.5][1.75,.75];
		{\Under[4,1.5][5.8,.75];}
		\Under[.2,.75][2,0];
		\Pole[.45][0,\TOP]
		\Over[3,1.5][.2,.75];
		{\Pole[5.55][0,\TOP]}
		\Over[5,1.5][4,.75];
		\Over[1.75,.75][1,0];
		\Over[4,.75][4,0];
		{\Over[5.8,.75][3,0];}
		\Caps[.45,5.55][0,\TOP][5]}$
	&\begin{tabular}{c}(twist \& wrap \\relations) \end{tabular}
	& $\TikZ{[diagram, scale=.5]
		\Frame[0,1.5][5]
		\node[wV] (L1) at (.5,.5) {}; 
		\node[wV] (L2) at (.5,1) {}; 
		\node[wV] (R1) at (4.5,.5) {}; 
		\node[wV] (R2) at (4.5,1) {}; 
		\draw[bend right=50] 
			(L2) to (t1) (b1) to (L1) (b5) to (b4) 
			(t5) to (R2) (R1) to (b3) (t2) to (t3);
		\draw (t4) to (b2); 
		}$\\\hline
\end{tabular}}

\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 $\delta, \delta_0, \delta_k \in \CC$. The \emph{two-boundary Temperley-Lieb algebra $TL^{(2)}_k$} 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, $TL^{(2)}_k$ 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}
$$
\TikZ{[diagram]
\draw[line width=1.5pt, dalert] (.5,-.3)--(.5,1.85)  (4.5,-.3)--(4.5,1.85);
\draw[densely dotted, dalert] (.5,-.3)--(4.5,-.3) (.5,1.85) --(4.5,1.85);
\foreach \x in {1,...,5}{\node[V] (b\x) at (\x*.75+.25, -.25) {}; \node[V] (t\x) at (\x*.75+.25, 1.75){};}
\foreach \y in {1,2,4,5}{\node[wV] (l\y) at (.5, \y*.25){}; \node[wV] (r\y) at (4.5, \y*.25){}; \draw (l\y)--(r\y);}
\node[wV] (r6) at (4.5, 1.55){}; \node[wV] (r0) at (4.5, -.05){}; 
\draw[bend right=60] 
		(t1) to  (t2)	(t3) to  (t4)	(b4) to  (b3)	(b2) to  (b1) ;
\draw[bend right=40] 	(t5) to (r6) (r0) to (b5);
\foreach \x in {-1,0,1}{
		\node[draw, fill =black, circle, inner sep=0pt, minimum size=.25pt] at (2.5, .75+.125*\x) {};}
\draw[|-|] (0,.1) to node[left, midway] {$2\ell$} (0,1.4);
} \in TL^{(2)}_k$$
So unlike the classical T-L algebras, $TL^{(2)}_k$ 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 $TL^{(2)}_k$: 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 $TL^{(2)}_k$: 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}\small
\def\SCALE{.5}
\alert{(1)} The \alert{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->{
\alert{(2)} Fix constants $q_0, q_{k},q\only<3->{=q_1=q_2=\cdots=q_{k-1}} \in \CC$. \\
The \alert{affine type C Hecke algebra}  $\cH_k$ is the quotient of $\CC\cB_k$ by the relations \only<3->{$(T_i - q_i^{1/2})(T_i + q_i^{-1/2}) = 0$.}\only<1-2>{$$(T_0 - q_0)(T_0 + q_0^{-1})=0, \quad (T_k - q_k)(T_k + q_k^{-1})=0$$
$$\text{and } \quad (T_i - q)(T_i + q^{-1}) = 0 \quad \text{for $i=1, \dots, k-1$.}$$}}

\medskip

\uncover<3->{
\alert{(3)} Set
\begin{align*}
\TikZ{[diagram]
	\draw[line width=1.5pt, alert] (.5,0) to (.5,1.25);
	\draw[densely dotted, alert] (.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); 
	}
	&\ =\     q_0\ \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 =q_0 - T_0)\\
\TikZ{[diagram]
	\draw[line width=1.5pt, alert] (1.5,0) to (1.5,1.25);
	\draw[densely dotted, alert] (.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); 
	}
	&\ =\     q_k\ \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 =q_k - T_k)\\
\TikZ{[diagram]
	\draw[densely dotted, alert] (.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); 
	}
	&\ = \ q \ \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[1,1][1.75,0]
			\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 =q - T_i)
\end{align*}\vspace{-.1in}

so that $e_j^2 = z_j e_j$ (for good $z_j$).}\\
\uncover<4->{The {\color{alert}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}{One advantage of using braids:}
\centerline{\begin{minipage}{6.5cm}
\def\TOP{1.5}
 \smallskip
\centerline{$\displaystyle
\TikZ{[line width=.75pt]
	\Under[4,1.5][5.8,.75];
	\cPole[5.55][0,\TOP][TypeA]
	\Over[5.8,.75][3,0];
	\draw[over] (2,1.5).. controls (2,1) and (4,1) .. (5,0);
	\Under[1,1.5][1,.75];
	\Under[.2,.75][2,0];
	\cPole[.45][0,\TOP][Univ]
	\Over[3,1.5][.2,.75];
	\Over[5,1.5][4,.75];
	\Over[1,.75][1,0];
	\Over[4,.75][4,0];
	\foreach \x in {1,...,5}{
		\draw (\x,0)  to (\x,-.05) node[bV, label=below:{$V$}]{};
		\draw (\x,\TOP)  to (\x,\TOP+.05) node[bV, label=above:{$V$}]{};
		\ifnum \x<5 
			\node at (\x+.5,1.86) {$\otimes$};
			\node at (\x+.5,-.4) {$\otimes$}; \fi
		}
	\node at (.55,\TOP+.38) {${\color{Univ}M} \!\otimes$};
	\node at (.55,-.4) {${\color{Univ}M} \!\otimes$};
	\node at (5.45,\TOP+.36) {$\otimes {\color{TypeA}N}$};
	\node at (5.45,-.4) {$\otimes {\color{TypeA}N}$};
	\cCaps[.45/Univ,5.55/TypeA][0,\TOP][5]
	}
$} \smallskip\pause
$$\def\arraystretch{.8}\begin{matrix}
		\text{\footnotesize \sl Move both poles}\\
		\text{\footnotesize \sl to the left } \end{matrix}
	\downarrow 
	\phantom{\begin{matrix}
		\text{\footnotesize \sl Move both poles}\\
		\text{\footnotesize \sl to the left } \end{matrix}}$$
\def\TOP{2}
 \smallskip
\centerline{$\displaystyle
\TikZ{[line width=.75pt]
	\draw (4,\TOP).. controls (3,.5) and (0,1).. (0,.5);
	\cPole[.3][0,\TOP][TypeA]
	\Over[0,.5][3,0];
	\draw[over] (2,\TOP) to (5,0);
	\draw[over] (1,\TOP) to (1,1);
	\Over[-.6,1.5][2,.65]; \draw[over] (2,.65) to (2,0);
	\cPole[-.4][0,\TOP][Univ]
	\Over[3,\TOP][-.6,1.5];
	\draw[over](5,\TOP) to (4,0);
%	\Over[5,\TOP][4,.75];
	\draw[over] (1,1.2) to (1,0);
%	\Over[4,.75][4,0];
	\foreach \x in {1,...,5}{
		\draw (\x,0)  to (\x,-.05) node[bV, label=below:{{$V$}}]{};
		\draw (\x,\TOP)  to (\x,\TOP+.05) node[bV, label=above:{{$V$}}]{};
		\ifnum \x<5 
			\node at (\x+.5,\TOP + .36) {{$\otimes$}};
			\node at (\x+.5,-.4) {{$\otimes$}}; \fi
		}
	\node at (-.28,\TOP + .38) {${\color{Univ}M} \otimes$};
	\node at (-.28,-.4) {${\color{Univ}M}\otimes$};
	\node at (.45,\TOP + .38) {$ {\color{TypeA}N}\otimes$};
	\node at (.45,-.4) {$ {\color{TypeA}N}\otimes$};
	\cCaps[.3/TypeA,-.4/Univ][0,\TOP][5]
	}
$}
\end{minipage}\pause$\TikZ{\draw (0,8) to (0,0);}$
\begin{minipage}{5.75cm}
Jucys-Murphy elements:
\def\TOP{2}\def\K{6}
$$Z_i = \TikZ{[scale=.5, line width=.75pt]
		\cPole[-.4][0,\TOP][Univ]
		\cPole[.3][0,\TOP][TypeA]
		 \foreach \x in {1,2} {\draw[over] (\x,0) -- (\x,1);}
		\Over[3,0][-.8,1]
		\Under[-.8,1][3,2]
		\cPole[-.4][1,2][Univ]
		\cPole[.3][1,2][TypeA]
		\foreach \x in {1,2} { \draw[over] (\x,1) -- (\x,2); }
		\foreach \x in {4,...,\K} {\draw[over] (\x,0) -- (\x,\TOP);}
		\cCaps[.3/TypeA,-.4/Univ][0,\TOP][\K]
		\Label[0,\TOP][3][{\footnotesize $i$}]
}
$$\pause
\begin{enumerate}[{\color{black}\footnotesize$\blacktriangleright$}]
\item Pairwise commute \pause
\item $Z(\cH_k)$ is {\small(type-C)} symmetric Laurent polynomials in $Z_i$'s \pause
\item Central characters indexed by $\cc \in \CC^k$ {\scriptsize(modulo signed permutations)} 
\end{enumerate}
\end{minipage}}
\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 $q_0$ and $q_k$: $r_1 = \log_{q}(q_0/q_k)$, $r_2 = \log_{q}(q_0q_k)$.}}
\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);
		\begin{scope}%noncalibrated
	\clip	(0,3)--(0,0)--(3,3)--(0,3);
		\uncover<2->{
		\draw[thick, fill=red] (0,.4) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (.2,.2) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (0,2.4) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (0,.6) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (.6,.6) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (2.4,2.4) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (1.5,1.5) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (0,1.7) circle (1.5pt) node[above]{};}
	\end{scope}
\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 $q_0$ and $q_k$: $r_1 = \log_{{q}}({q_0/q_k})$, $r_2 = \log_{{q}}({q_0q_k})$.}
\end{frame}


\begin{frame}{A {\small little} more detail}
\small \begin{enumerate}[$\bullet$]
\item $J$ is determined by a set of positive roots {\small (corresp.\ to hyperplanes)}.\\
\item For ``nice'' characters, there is a bijection between alcoves and marked type-C generalized Young tableaux.
\item ``Intertwining operators'' $\tau_i$ move between alcoves; \\
$\qquad$\hfill dotted lines correspond to $\tau_i = 0$. 
\end{enumerate}
\normalsize

\only<1>{\centerline{\begin{tikzpicture}[scale=1.4]
			\draw[black!25] (-45:-1.2)--(-45:1.2) ;
			\draw[dotted, thick] (45:-2)--(45:2);
			\draw[dotted, thick] (-2,0)--(2,0);
			\draw[black!25]  (0,-1.2)--(0,1.2);
			\node at  (112.5 - 45:.9) {\tiny$1$};
			\node at  (112.5 - 45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{1}\BOX{1,-2}{2}  
					\BOX{0,0}{-1} \BOX{-1,0}{-2} 
					\DOT{1,-2} \DOT{0,1}
				\end{tikzpicture}};
			\node at  (112.5 - 2*45:.9) {\tiny$s_1$};
			\node at  (112.5 - 2*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{2}\BOX{0,-3}{1}  
					\BOX{-2,-2}{-2} \BOX{-2,-1}{-1} 
					\DOT{0,-3} \DOT{-1,0}
				\end{tikzpicture}};
			\node at  (112.5 - 3*45:.9) {\tiny$s_0 s_1$};
			\node at  (112.5 - 3*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
				\BOX{0,-2}{2}\BOX{0,-3}{-1}  
				\BOX{-2,-2}{-2} \BOX{-2,-1}{1} 
				\DOT{1,-2} \DOT{-2,-1}
			\end{tikzpicture}};
			\node at  (112.5 - 4*45:.9) {\tiny$~s_1s_0s_1$};
			\node at  (112.5 - 4*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{1}\BOX{0,-3}{-2}  
					\BOX{-2,-2}{-1} \BOX{-2,-1}{2} 
					\DOT{1,-2} \DOT{-2,-1}
				\end{tikzpicture}};
			\node at  (67.5 + 1*45:.9) {\tiny$s_0$}; 
			\node at  (67.5 + 1*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{-1}\BOX{1,-2}{2}  
					\BOX{0,0}{1} \BOX{-1,0}{-2} 
					\DOT{1,-2} \DOT{0,1}
				\end{tikzpicture}}; 
			\node at  (67.5 + 2*45:.9) {\tiny$s_1s_0$}; 
			\node at  (67.5 + 2*45:1.5) {
					\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{-2}\BOX{1,-2}{1}  
					\BOX{0,0}{2} \BOX{-1,0}{-1} 
					\DOT{1,-2} \DOT{0,1}
				\end{tikzpicture}
			}; 
			\node at  (67.5 + 3*45:.9) {\tiny$s_0s_1s_0$}; 
			\node at  (67.5 + 3*45:1.65) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{-2}\BOX{1,-2}{-1}  
					\BOX{0,0}{2} \BOX{-1,0}{1} 
					\DOT{2,-1} \DOT{-1,0}
				\end{tikzpicture} }; 
			\node at  (67.5 + 4*45:.9) {\tiny$s_1s_0s_1s_0~$}; 
			\node at  (67.5 + 4*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{-1}\BOX{0,-3}{-2}  
					\BOX{-2,-2}{1} \BOX{-2,-1}{2} 
					\DOT{1,-2} \DOT{-2,-1}
				\end{tikzpicture}}; 
			%%%%%%%%%%%%%%%%%%
			\node[B] at (0,0){};
			\foreach \x [count=\c from 1] in {1, ..., 9}{\node[V] (a\x) at (112.5 - \c*45:.5){}; }
			\foreach \x [count=\c from 4] in { 3, 4}{ \draw (a\x)--(a\c);}
			\foreach \x [count=\c from 8] in {7,8}{\draw (a\x)--(a\c);}
			\node at  (90+40:2) {\small$J = \emptyset$};
			\node[above] at  (0:2.5) {\small$J = \{\vep_2-\vep_1\}$};
			\node[below] at  (185:2.2) {\small$J = \{\vep_2\}$};
			\node at  (-90+22.5:2.2) {\small$J = \{\vep_2, \vep_2-\vep_1\}$};
		\end{tikzpicture}}}\only<2>{\centerline{\begin{tikzpicture}[scale=1.4]
			\draw[black!25] (-45:-1.2)--(-45:1.2) ;
			\draw[dotted, thick] (45:-2)--(45:2);
			\draw[dotted, thick] (-2,0)--(2,0);
			\draw[black!25]  (0,-1.2)--(0,1.2);
			\node at  (112.5 - 45:.9) {\tiny$1$};
			\node at  (112.5 - 45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{1}\cBOX{yellow!50}{1,-2}{2}  
					\BOX{0,0}{-1} \cBOX{blue!30}{-1,0}{-2} 
					\DOT{1,-2} \DOT{0,1}
				\end{tikzpicture}};
			\node at  (112.5 - 2*45:.9) {\tiny$s_1$};
			\node at  (112.5 - 2*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\cBOX{yellow!50}{0,-2}{2}\cBOX{yellow!50}{0,-3}{1}  
					\cBOX{blue!30}{-2,-2}{-2} \cBOX{blue!30}{-2,-1}{-1} 
					\DOT{0,-3} \DOT{-1,0}
				\end{tikzpicture}};
			\node at  (112.5 - 3*45:.9) {\tiny$s_0 s_1$};
			\node at  (112.5 - 3*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
				\BOX{0,-2}{2}\cBOX{blue!30}{0,-3}{-1}  
				\BOX{-2,-2}{-2} \cBOX{yellow!50}{-2,-1}{1} 
				\DOT{1,-2} \DOT{-2,-1}
			\end{tikzpicture}};
			\node at  (112.5 - 4*45:.9) {\tiny$~s_1s_0s_1$};
			\node at  (112.5 - 4*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{1}\cBOX{blue!30}{0,-3}{-2}  
					\BOX{-2,-2}{-1} \cBOX{yellow!50}{-2,-1}{2} 
					\DOT{1,-2} \DOT{-2,-1}
				\end{tikzpicture}};
			\node at  (67.5 + 1*45:.9) {\tiny$s_0$}; 
			\node at  (67.5 + 1*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{-1}\cBOX{yellow!50}{1,-2}{2}  
					\BOX{0,0}{1} \cBOX{blue!30}{-1,0}{-2} 
					\DOT{1,-2} \DOT{0,1}
				\end{tikzpicture}}; 
			\node at  (67.5 + 2*45:.9) {\tiny$s_1s_0$}; 
			\node at  (67.5 + 2*45:1.5) {
					\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{-2}\cBOX{yellow!50}{1,-2}{1}  
					\BOX{0,0}{2} \cBOX{blue!30}{-1,0}{-1} 
					\DOT{1,-2} \DOT{0,1}
				\end{tikzpicture}
			}; 
			\node at  (67.5 + 3*45:.9) {\tiny$s_0s_1s_0$}; 
			\node at  (67.5 + 3*45:1.65) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\cBOX{blue!30}{0,-2}{-2}\cBOX{blue!30}{1,-2}{-1}  
					\cBOX{yellow!50}{0,0}{2} \cBOX{yellow!50}{-1,0}{1} 
					\DOT{2,-1} \DOT{-1,0}
				\end{tikzpicture} }; 
			\node at  (67.5 + 4*45:.9) {\tiny$s_1s_0s_1s_0~$}; 
			\node at  (67.5 + 4*45:1.5) {
				\begin{tikzpicture}[xscale = .3, yscale=-.3, every node/.style={ inner sep=1pt}]
					\BOX{0,-2}{-1}\cBOX{blue!30}{0,-3}{-2}  
					\BOX{-2,-2}{1} \cBOX{yellow!50}{-2,-1}{2} 
					\DOT{1,-2} \DOT{-2,-1}
				\end{tikzpicture}}; 
			%%%%%%%%%%%%%%%%%%
			\node[B] at (0,0){};
			\foreach \x [count=\c from 1] in {1, ..., 9}{\node[V] (a\x) at (112.5 - \c*45:.5){}; }
			\foreach \x [count=\c from 4] in { 3, 4}{ \draw (a\x)--(a\c);}
			\foreach \x [count=\c from 8] in {7,8}{\draw (a\x)--(a\c);}
			\node at  (90+40:2) {\small$J = \emptyset$};
			\node[above] at  (0:2.5) {\small$J = \{\vep_2-\vep_1\}$};
			\node[below] at  (185:2.2) {\small$J = \{\vep_2\}$};
			\node at  (-90+22.5:2.2) {\small$J = \{\vep_2, \vep_2-\vep_1\}$};
		\end{tikzpicture}}}
		\end{frame}



%%%%%%%%%%
%%%%%%%%%%
\begin{frame}
\centerline{$\TikZ{[scale=1.875]
	\filldraw[gray!20] (0,3)--(0,0)--(3,3)--(0,3);
	\begin{scope}%noncalibrated
	\clip	(0,3)--(0,0)--(3,3)--(0,3);
		\only<1,3>{\draw[thick, fill=red] (0,.4) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (.2,.2) circle (1.5pt) node[above]{};}
		%
		\only<1>
		{\draw[thick, fill=red] (0,2.4) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (0,.6) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (.6,.6) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (2.4,2.4) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (1.5,1.5) circle (1.5pt) node[above]{};
		\draw[thick, fill=red] (0,1.7) circle (1.5pt) node[above]{};}
	\end{scope}	
\draw[thick] (0,-.2)--(0,3) node[above] {$\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]{};
%	\draw[thick, fill=red] (0,.4) circle (1pt) node[above]{};
%	\draw[thick, fill=red] (.2,.2) 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>{
%		\draw[thick, fill=red] (0,2.4) circle (1pt) node[above]{};
%		\draw[thick, fill=red] (0,.6) circle (1pt) node[above]{};
%		\draw[thick, fill=red] (.6,.6) circle (1pt) node[above]{};
%		\draw[thick, fill=red] (2.4,2.4) circle (1pt) node[above]{};
%		\draw[thick, fill=red] (1.5,1.5) circle (1pt) node[above]{};
%		\draw[thick, fill=red] (0,1.7) circle (1pt) node[above]{};
		\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 representations of $\cH_k$ that factor through the Temperley-Lieb quotient are as 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]
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%%%%%%
\hspace{-.75cm}
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	\draw[alert, thick, <->] (nR7) to node[alert, above] {\tiny$\tau_0$} (nL8);}
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	\draw[<->] (nR8) to node[ right] {\tiny$\tau_0$} (n9);}
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$$
\end{frame}




\begin{frame}{See also...}

Specific:
\begin{enumerate}[\qquad]
\item[{[}DR{]}] Daugherty and Ram, ``Two boundary Hecke Algebras and combinatorics of type C'', arXiv:1804.10296. \\
\item[{[}GN{]}] de Gier and Nichols, ``The two-boundary Temperley-Lieb algebra'', J.\ Algebra 321 (2009), no.\ 4, 1132--1167. 
\end{enumerate}

\medskip

General:
\begin{enumerate}[\qquad]
\item[{[}BR{]}] Barcelo and Ram, ``Combinatorial representation theory,'' New perspectives in algebraic combinatorics (Berkeley, CA, 1996--97), 23--90,
Math.\ Sci.\ Res.\ Inst.\ Publ., 38, Cambridge Univ. Press, Cambridge, 1999.
\end{enumerate}

\medskip

\begin{center}
https://zdaugherty.ccnysites.cuny.edu\\
~\\
\alert{Thanks!}
\end{center}


\end{frame}
\end{document}