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 \def\ANSWER[#1]{\uncover<2->{{\color{dblue} #1}}}
\title{Centralizer properties of \\
the affine Hecke algebra of type
C}
\author[Zajj Daugherty]{Zajj Daugherty\\
(joint with Arun Ram)}
\institute{Dartmouth College}
\date{January 11, 2013}
\begin{document}

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


\begin{frame}{}
The \alert{two-boundary braid group} is the group $\cB_k$ generated by 
$ T_0,  T_1,\ldots,  T_k$,
with relations \begin{equation*}\label{Bdefrels}
\begin{tikzpicture}[line width=1pt]
	\foreach \x in {0, 1, 2, 4,5,6}{
		\draw (\x,0) circle (2.5pt);
		\node(\x) at (\x,0){};
		}
	\foreach \x in {0, 1, 2}
		{\node[label=above:$ 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) (4)--(5);
	\draw[dashed] (2)--(4);
	\only<3>{
		\draw[dgreen] (1,0) circle (2.5pt);
		\draw[dgreen] (2,0) circle (2.5pt);
		\draw[dgreen] (1)--(2);
	}
	\only<4>{
		\draw[dgreen] (1,0) circle (2.5pt);
		\draw[dgreen] (0,0) circle (2.5pt);
		\draw[dgreen, double distance = 2pt] (1)--(0);
	}
\end{tikzpicture}\end{equation*}


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

\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}{}
The \alert{two-boundary braid group} is the group $\cB_k$ generated by 
$ T_0,  T_1,\ldots,  T_k$,
with relations
\begin{equation*}\label{Bdefrels}
\begin{tikzpicture}[line width=1pt]
	\foreach \x in {0, 1, 2, 4,5,6}{
		\draw (\x,0) circle (2.5pt);
		\node(\x) at (\x,0){};
		}
	\foreach \x in {0, 1, 2}
		{\node[label=above:$ 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) (4)--(5);
	\draw[dashed] (2)--(4);
\end{tikzpicture}\end{equation*}
\hrule

\vfill

\alert{\bf Two (isomorphic) quotients, two perspectives:}\\

\vfill\pause
\alert{1. } Fix $ a_1, a_2, b_1, b_2, q \in \CC^\times$. The \alert{affine Hecke algebra $\cH_k$ of type C} is the quotient of $\CC\cB_k$ by
$$\hspace{-.25in}\alert{(*)} \quad 
( T_0 - a_1) ( T_0 - a_2) = 0, \quad 
( T_k - b_1) ( T_k - b_2) = 0, \quad
( T_i - q) ( T_i + q^{-1}) = 0.$$
{\small (various assumptions are made about  $a_i, b_i$ depending on context)}
\pause


\alert{2. } Let $A, B, C$ be finite dim'l $U_q \fg$-modules. Then $\CC\cB_k$ acts on 
$$B \otimes \underbrace{C \otimes \cdots \otimes C}_{k \text{ factors}} \otimes A$$
Under good (to be defined) conditions, this action factors through the quotient $\alert{(*)}$.
\end{frame}


\begin{frame}{Representation theory of $\cH_k$}
The representations of $\cH_k$ are indexed by \alert{skew local regions}. For example, when $k=2$:

\uncover<2->{
\centerline{
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\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] {$\fh^{\alpha_1+\alpha_2}$};
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	\draw[dotted, thick, dgrey] (-3+.4,3)--(3,-3+.4) node[right]{\tiny$c_2=-c_1+1$};
	\draw[dotted, thick, dgrey] (-3,3-.4)--(3-.4,-3) node[below right]{\tiny$c_2=-c_1-1$};
	\draw[dotted, thick, dgrey] (3,3-.4)--(-3+.4,-3) node[below left]{\tiny$c_2=c_1-1$};}
\uncover<6->{
	\draw[dashed] (3,.8)--(-3,.8) node[left] {\tiny$c_2=\gamma_1$};
	\draw[dashed] (3,2.2)--(-3,2.2) node[left] {\tiny$c_2=\gamma_2$};
	\draw[dashed] (.8,-3)--(.8,3) node[above] {\tiny$c_1=\gamma_1$};
	\draw[dashed] (2.2,-3)--(2.2,3) node[above] {\tiny$c_1=\gamma_2$};
	}
\uncover<7->{
	\draw[dashed] (3,-.8)--(-3,-.8) node[left] {\tiny$c_2=-\gamma_1$};
	\draw[dashed] (3,-2.2)--(-3,-2.2) node[left] {\tiny$c_2=-\gamma_2$};
	\draw[dashed] (-.8,3)--(-.8,-3.2) node[below] {\tiny$c_1=-\gamma_1$};
	\draw[dashed] (-2.2,3)--(-2.2,-3.2) node[below] {\tiny$c_1=-\gamma_2$};
}
\end{tikzpicture}}}
\uncover<3->{\vspace{-.09in} \scriptsize Points correspond to central characters. \uncover<6->{The $\gamma_i$s depend on $\cH_k$'s parameters.}}
\end{frame}


\begin{frame}{Representation theory of $\cH_k$}
Classically, we index the representations of $\cH_k$ by \alert{skew local regions}. For example, when $k=2$:

\centerline{
\begin{tikzpicture}[scale=1.875]
	\filldraw[gray!20] (0,3)--(0,0)--(3,3)--(0,3);
\draw[thick] (0,-.2)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-.2,-.2)--(3,3) node[above right] {$\fh^{\alpha_1}$};
\draw[gray] (-.2,0)--(3,0) node[right] {\color{gray}$\fh^{\alpha_2+2\alpha_1}$};
	\draw[dotted, thick, black!70] (3-.4,3)--(-.2,-.2+.4) node[left]{\tiny$c_2=c_1+1$};
	\draw[dotted, thick, black!70] (3,.6)--(-.2,.6) node[left] {\tiny$c_2=\gamma_1$};
	\draw[dotted, thick, black!70] (3,2.4)--(-.2,2.4) node[left] {\tiny$c_2=\gamma_2$};
	\draw[dotted, thick, black!70] (.6,-.2)--(.6,3) node[above] {\tiny$c_1=\gamma_1$};
	\draw[dotted, thick, black!70] (2.4,-.2)--(2.4,3) node[above] {\tiny$c_1=\gamma_2$};
\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[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);
			\filldraw[yellow!70] (.6-.15,1-.15)--(.6,1-.2)--(.6,1);
			\filldraw[red!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[red!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[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+.2,2.4)--(.6+.15,2.4+.15)--(.6,2.4+.2);
		\REG[.6,2.4]}
\uncover<2->{
	% 2
	\filldraw[dgreen] (.6,2.4) circle (1pt) node[above]{};
	% 3/1
	\filldraw[dgreen] (.2,.6) circle (1pt) node[above]{};
	\filldraw[dgreen] (.6,1) circle (1pt) node[above]{};
	\filldraw[dgreen] (2,2.4) circle (1pt) node[above]{};
	\filldraw[dgreen] (2.4,2.8) circle (1pt) node[above]{};
	%4:
	\filldraw[dgreen] (.6,1.7) circle (1pt) node[above]{};
	\filldraw[dgreen] (1.3,1.7) circle (1pt) node[above]{};
	\filldraw[dgreen] (1.3,2.4) circle (1pt) node[above]{};
	%8
	\filldraw[dgreen] (1,2) circle (1pt) node[above]{};
	}
\end{tikzpicture}}
\centerline{\scriptsize(points correspond to central characters)}
\end{frame}

\begin{frame}{Centralizer properties}
Let $U = U_q \fg$ be the quantum group for a finite dim'l reductive Lie algebra. 
We're interested in certain finite dimensional simple $U$-modules $\alert{L(\lambda)}$ indexed by  \alert{partitions:}
$$\lambda = \begin{picture}(40, 23)(0,2)
		\multiput(0,0)(10,0){4} {\line(0,1){20}} %V
		\multiput(0,0)(0,10){3} {\line(1,0){30}} %H
		%
		\put(30,20) {\line(1,0){10}} %H
		\put(30,10) {\line(1,0){10}} %H
		%
		\put(40,20) {\line(0,-1){10}} %V
		%
		\put(0,-10) {\line(1,0){10}} %H
		%
		\put(0,0) {\line(0,-1){10}} %V
		\put(10,0) {\line(0,-1){10}} %V
		\uncover<4->{
		\put(3,12){\text{\footnotesize\alert{0}}}
		\put(1,2){\text{\footnotesize\alert{-1}}}
		\put(1,-8){\text{\footnotesize\alert{-2}}}
		%
		\put(13,12){\text{\footnotesize\alert{1}}}
		\put(13,2){\text{\footnotesize\alert{0}}}
		%
		\put(23,12){\text{\footnotesize\alert{2}}}
		\put(23,2){\text{\footnotesize\alert{1}}}
		%
		\put(33,12){\text{\footnotesize\alert{3}}}
}
		\end{picture}
$$
~
\centerline{\small (drawn as a collection of boxes piled up and to the left)}

\pause
\vfill

In particular, rectangular partitions:
$$
	(a^c) = \begin{picture}(60, 18)(-10,12)
		\multiput(10,2)(10,0){3} {\line(0,1){26}} %V
		\multiput(2,10)(0,10){2} {\line(1,0){36}} %H
		%
		\thicklines
		\uncover<5->{\alert{
		\put(40,20){\dashbox{1}(10,10){\footnotesize $a$}}
		\put(0,-10){\dashbox{1}(10,10){\footnotesize -$c$}}
		}}		
		\multiput(0,0)(40,0){2} {\line(0,1){30}} %V
		\multiput(0,0)(0,30){2} {\line(1,0){40}} %H
		%
		\put(-10,12){$c$}
		\put(17,32){$a$}
		%
     \end{picture} 
     $$
     ~
     
     \pause
 If a box is in column $i$ and row $j$ of a partition $\lambda$, then the \emph{content} of that box is \\
	\centerline{$c(\text{box}) = i-j.$}%

\end{frame}


\begin{frame}{Centralizer properties}
\begin{thm}[\small D.-Ram]
\begin{enumerate}
\item Let $U = U_q \fg$, and let $A$, $B$, and $C$ be finite dim'l $U$-modules.  The two-boundary braid group $\cB_k$ acts on $B \otimes (C)^{\otimes k} \otimes A$
(via $R$-matrices) and this action commutes with that of $U$. \pause

\centerline{\def\K{5}
\def\TOP{1}
$\begin{tikzpicture}[yscale=.8,line width=1pt ]
\SymPolesTop[\TOP][\K]
\SymPolesBottom[0][\K]
\draw[style=over] (1,\TOP)--(3,0);
\Over[3,\TOP][.15, .3*\TOP]
\RPoleO[0,\TOP]
\Over[.15, .3*\TOP][1,0]
\draw[style=over] (4,\TOP)--(2,0); 
\Under[5,\TOP][5.85,.5*\TOP] 
\RRPoleO[0,\TOP][\K]
\Over[5.85,.5*\TOP][5,0]
\draw[style=over](2,\TOP)--(4,0);
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,\TOP) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
  \node at (.415, \TOP+.5) {\scriptsize $B$};
  \node at (.415, -.5) {\scriptsize $B$};
    \node at (5.565, \TOP+.5) {\scriptsize $A$};
  \node at (5.565, -.5) {\scriptsize $A$};
  \node at (.715, \TOP+.5) {\scriptsize $\otimes$};
  \node at (.715, -.5) {\scriptsize $\otimes$};
  \node at (5.3, \TOP+.5) {\scriptsize $\otimes$};
  \node at (5.3, -.5) {\scriptsize $\otimes$};
  \foreach \x in {1,...,5}{
 	\node at (\x, \TOP+.5) {\scriptsize $C$};
 	\node at (\x, -.5) {\scriptsize $C$};
  }
  \foreach \x in {1,...,4}{
 	\node at (\x+.5, \TOP+.5) {\scriptsize $\otimes$};
 	\node at (\x+.5, -.5) {\scriptsize $\otimes$};
  }
\end{tikzpicture}$}
\end{enumerate}
\end{thm}
\pause
$R$-matrices: $U$ has an associated invertible element $R=\sum_{\cR} R_1\otimes R_2$ of $U \otimes U$ that gives us a map\\
\centerline{\begin{tikzpicture}[xscale=.8, yscale=.6, line width=1pt ]
	\node at (-3,1.4) 
		{$\check R_{MN}\colon M\otimes N \longrightarrow N\otimes M$};
	\Under[1,1][2,2] 	\Over[1,2][2,1]
	\Nodes[1][2]	\Nodes[2][2]
	\node at (1.5,2.5) {$M \otimes N$};
	\node at (1.5,.5) {$N \otimes M$};
\end{tikzpicture}}
This map acts a component $L(\lambda)$ of $L(\mu) \otimes L(\begin{tikzpicture}[scale=.25]\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);\end{tikzpicture})$ by $q^{c(\lambda/\mu)}$. 

\end{frame}


\begin{frame}{Centralizer properties}
\begin{thm}[\small D.-Ram]
\begin{enumerate}
\item[2.] If $\fg = \fgl_n$, $A= L((a^c))$, $B = L((b^d))$, and $C = L(\begin{tikzpicture}[scale=.25]\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);\end{tikzpicture})$, then the action in \alert{1.} factors through the quotient by
\end{enumerate}
$$\hspace{-.2in}(T_0 - q^{2b})(T_0-q^{-2d}) = 0, ~ (T_k - q^{2a})(T_k-q^{-2c}) = 0, ~ (T_i - q)(T_i+q^{-1})=0.$$
\end{thm}
\pause
\vfill

\centerline{\def\TOP{1}\def\K{1}
\begin{tikzpicture}[line width=1pt , scale=.75]
\node[left] at (.3,.5*\TOP) {$T_0: $};
%%%%%%%%%%%%%%
\pgftransformxshift{0cm}
%%%%%%%%%%%%%
	\PoleTop[\TOP]
	\PoleBottom[0]
	\Under[.2,.5][1,1.2]
	\RPoleO[0,1]
	\Over[1,-.2][.2,.5]
	\Nodes[-.2][\K]
	\Nodes[\TOP+.2][\K]
\Label[0-.3,\TOP+.3][.7][\footnotesize $B\!\otimes\!C$]
%%%%%%%%%%%%%
\pgftransformxshift{3.75 cm}
%%%%%%%%%%%%%
\node[left] at (1,.5*\TOP) {$\cdots \qquad T_k:  \cdots$};
%%%%%%%%%%%%%
%\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
%%%%%%%%%%%%%%
\draw (1+.5,\TOP) -- (1+.5,\TOP+.25) (1+.65,\TOP) -- (1+.65,\TOP+.25) ;
\draw (1+.65,\TOP+.25) arc (0:360:2pt and 1.5pt);
\draw (1+.5,0) -- (1+.5,-.25) (1+.65,0) -- (1+.65,-.25) ;
\draw (1+.65,-.25)  arc (0:-180:2pt and 1.5pt);
\Under[\K,-.2][\K+.8,.5]
\RRPoleO[0,1][\K]
\Over[\K+.8,.5][\K,1.2]
\Nodes[0-.2][\K]
\Nodes[\TOP+.2][\K]
\Label[0-.3,\TOP+.3][1.3][\footnotesize $C\!\otimes\!A$]
\pgftransformxshift{3.5 cm}
%%%%%%%%%%%%%
\node[left] at (1,.5*\TOP) {$T_i:$};
%%%%%%%%%%%%%%
%\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%%
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[0,\TOP][1.5][\footnotesize $C \otimes C$]
\end{tikzpicture}
}

$$\qquad
	\begin{picture}(60, 18)(-10,12)
		\multiput(10,12)(10,0){2} {\line(0,1){16}} %V
		\multiput(2,20)(0,10){1} {\line(1,0){26}} %H
		%
		\thicklines
		{\alert{
		\put(30,20){\dashbox{1}(10,10){\footnotesize $b$}}
		\put(0,0){\dashbox{1}(10,10){\footnotesize -$d$}}
		}}		
		\multiput(0,10)(30,0){2} {\line(0,1){20}} %V
		\multiput(0,10)(0,20){2} {\line(1,0){30}} %H
		%
		\put(-10,15){$d$}
		\put(15,32){$b$}
		%
     \end{picture} 
\qquad\quad
	\begin{picture}(60, 18)(-10,12)
		\multiput(10,2)(10,0){3} {\line(0,1){26}} %V
		\multiput(2,10)(0,10){2} {\line(1,0){36}} %H
		%
		\thicklines
		{\alert{
		\put(40,20){\dashbox{1}(10,10){\footnotesize $a$}}
		\put(0,-10){\dashbox{1}(10,10){\footnotesize -$c$}}
		}}		
		\multiput(0,0)(40,0){2} {\line(0,1){30}} %V
		\multiput(0,0)(0,30){2} {\line(1,0){40}} %H
		%
		\put(-10,12){$c$}
		\put(17,32){$a$}
		%
     \end{picture} 
     \qquad\quad
	\begin{picture}(60, 18)(-10,12)
		\thicklines
		{\alert{
		\put(10,20){\dashbox{1}(10,10){\footnotesize $1$}}
		\put(0,10){\dashbox{1}(10,10){\footnotesize -$1$}}
		}}		
		\multiput(0,20)(10,0){2} {\line(0,1){10}} %V
		\multiput(0,20)(0,10){2} {\line(1,0){10}} %H
     \end{picture} 
     $$



\end{frame}

\begin{frame}{Exploring our new favorite tensor space, $A \otimes B \otimes C^{\otimes k}$} 
Move the right pole to the left:
$$
{\def\TOP{1}\def\K{5}
\begin{tikzpicture}[scale=.8, line width = 1pt]
\SymPolesTop[\TOP][\K]
\SymPolesBottom[0][\K]
\RPole[0,\TOP]
\Under[\K,0][\K+1,.5*\TOP]
\RRPoleO[0,.5*\TOP][\K]
\RRPole[.5*\TOP,\TOP][\K]
\Over[\K+1,.5*\TOP][\K,\TOP]
 \foreach \x in {1,...,4} {
	 \draw (\x,0) -- (\x,\TOP);
	 }
\Nodes[0][\K]
\Nodes[\TOP][\K]
  \node at (.415, 1.6) {\scriptsize $B$};
  \node at (.415, -.5) {\scriptsize $B$};
    \node at (5.65, 1.6) {\scriptsize $A$};
  \node at (5.65, -.5) {\scriptsize $A$};
  \node at (.715, 1.6) {\scriptsize $\otimes$};
  \node at (.715, -.5) {\scriptsize $\otimes$};
  \node at (5.3, 1.6) {\scriptsize $\otimes$};
  \node at (5.3, -.5) {\scriptsize $\otimes$};
  \foreach \x in {1,...,5}{
 	\node at (\x, 1.6) {\scriptsize $C$};
 	\node at (\x, -.5) {\scriptsize $C$};
  }
  \foreach \x in {1,...,4}{
 	\node at (\x+.5, 1.6) {\scriptsize $\otimes$};
 	\node at (\x+.5, -.5) {\scriptsize $\otimes$};
  }
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
\pgftransformxshift{2.5 cm}
%%%%%%%%%%%%%
\node[right]  at (-1.5, .5*\TOP)  {$=$};
\PolesTop[\TOP]
\PolesBottom[0]
\LPole[0,1]
\Over[\K,0][-.5,.5*\TOP]
\Under[-.5,.5*\TOP][\K,\TOP]
\LPoleO[.5*\TOP,\TOP]
\RPoleO[0,\TOP]
 \foreach \x in {1,...,4}  {
	 \draw[thin, style=over] (\x,0) -- (\x,\TOP);
	 }
\Nodes[0][\K]
\Nodes[\TOP][\K]
  \node at (.415, 1.6) {\scriptsize $B$};
  \node at (.415, -.5) {\scriptsize $B$};
    \node at (-.2, 1.6) {\scriptsize $A$};
  \node at (-.2, -.5) {\scriptsize $A$};
  \node at (.715, 1.6) {\scriptsize $\otimes$};
  \node at (.715, -.5) {\scriptsize $\otimes$};
  \node at (.15, 1.6) {\scriptsize $\otimes$};
  \node at (.15, -.5) {\scriptsize $\otimes$};
  \foreach \x in {1,...,5}{
 	\node at (\x, 1.6) {\scriptsize $C$};
 	\node at (\x, -.5) {\scriptsize $C$};
  }
  \foreach \x in {1,...,4}{
 	\node at (\x+.5, 1.6) {\scriptsize $\otimes$};
 	\node at (\x+.5, -.5) {\scriptsize $\otimes$};
  }
\end{tikzpicture}
}
$$

\uncover<2->{
New favorite generators: 
$${\def\TOP{1}\def\K{1}
\begin{tikzpicture}[line width=1pt , scale=.75]
\node[left] at (0,.5*\TOP) {$T_0=$};
%%%%%%%%%%%%%
\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
	\PolesTop[\TOP]
	\PolesBottom[0]
	\LPoleO[0,1]
	\Under[.2,.5][1,1]
	\RPoleO[0,1]
	\Over[1,0][.2,.5]
	\Nodes[0][\K]
	\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{2.5 cm}
%%%%%%%%%%%%%
\node at (0,.5*\TOP) {$T_i=$};
%%%%%%%%%%%%%%
%\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%%
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[0,\TOP][1][\footnotesize $i$]
	\Label[0,\TOP][2][\footnotesize$i$+1]
%%%%%%%%%%%%%
\pgftransformxshift{6 cm}
\pgftransformxshift{.25 cm}
%%%%%%%%%%%%%
\node[left]  at (0, .5*\TOP)  {and \quad $Y_1 = $};
%%%%%%%%%%%%%
\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
\PolesTop[\TOP]
\PolesBottom[0]
\LPole[0,.5*\TOP]
\RPoleO[0,.5*\TOP]
\Over[1,0][-.5,.5*\TOP]
\Under[-.5,.5*\TOP][1,\TOP]
\LPoleO[.5*\TOP,\TOP]
\RPoleO[.5*\TOP,\TOP]
% \foreach \x in {2,...,\K}  {
%	 \draw[thin] (\x,0) -- (\x,\TOP);
%	 }
\Nodes[0][\K]
\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{\K cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {.};
\end{tikzpicture}}$$


\centerline{Let $Y_2 = T_1 Y_1 T_1 = \begin{minipage}{2.25cm}{\def\TOP{1}\def\K{2}
\begin{tikzpicture}[line width=1pt , scale=.75]
\PolesTop[\TOP]
\PolesBottom[0]
\LPole[0,.5*\TOP]
\RPoleO[0,.5*\TOP]
\Under[-.5,.5*\TOP][2,\TOP]
\LPoleO[.5*\TOP,\TOP]
\RPoleO[.5*\TOP,\TOP]
	 \draw[over] (1,0) -- (1,\TOP);
\Over[2,0][-.5,.5*\TOP]
\Nodes[0][\K]
\Nodes[\TOP][\K]\end{tikzpicture}}
\end{minipage}$.}}
\end{frame}


\begin{frame}{Exploring our new favorite tensor space, $A \otimes B \otimes C^{\otimes k}$} 
~\\
Let $A = L((a^c))$ and $B = L((b^d))$. 
Then 
	$$A \otimes B = \bigoplus_{\lambda \in \Lambda} L(\lambda) \qquad \text{\tiny (multiplicity one!)}$$

where $\Lambda$ is the following set of partitions\only<1-4>{:}\only<5>{\ldots} \\
{\footnotesize(Littlewood-Richardson, Okada)}

\only<1>{\phantom{$$
	\begin{picture}(90, 97)(0,0)
     \end{picture} 
$$}}\only<2>{$$
	\begin{picture}(90, 65)(0,0)
		%\thicklines
		\multiput(0,0)(50,0){2} {\line(0,1){40}} %V
		\multiput(0,0)(0,40){2} {\line(1,0){50}} %H
		%
		\put(-10,17){c}
		\put(22,42){a}
		%
		\multiput(50,15)(0,25){2} {\line(1,0){25}} %H
		\put(75,15) {\line(0,1){25}} %V
		%
		\put(77,27){d}
		\put(59,42){b}
     \end{picture} 
$$}\only<3>{$$
	\begin{picture}(90, 65)(0,0)
		%\thicklines
		\multiput(0,0)(50,0){2} {\line(0,1){40}} %V
		\multiput(0,0)(0,40){2} {\line(1,0){50}} %H
		%
%		\put(-10,17){p}
%		\put(22,42){a}
		%
		\multiput(50,15)(0,25){2} {\line(1,0){25}} %H
		\put(75,15) {\line(0,1){25}} %V
		%
%		\put(77,27){q}
%		\put(59,42){b}
		%
		\alert{
		\put(75,35){\line(-1,0){10}}
		\put(65,35){\line(0,-1){15}}
		\put(65,20){\line(-1,0){5}}
		\put(60,20){\line(0,-1){5}}
		}
     \end{picture} 
$$}\only<4>{$$
	\begin{picture}(90, 65)(0,0)
		%\thicklines
		\multiput(0,0)(50,0){2} {\line(0,1){40}} %V
		\multiput(0,0)(0,40){2} {\line(1,0){50}} %H
		%
		\put(50,40){\line(1,0){25}} %H
		\put(50,15){\line(1,0){10}} %H
		\put(75,40) {\line(0,-1){5}} %V
		%
		\alert{
		\put(75,35){\line(-1,0){10}}%H
		\put(65,35){\line(0,-1){15}} %V
		\put(65,20){\line(-1,0){5}}%H
		\put(60,20){\line(0,-1){5}} %V
		}
		%
		\put(0,0) {\line(0,-1){20}} %V
		%
		\alert{
		\put(0,-20){\line(1,0){10}} %H
		\put(10,-20){\line(0,1){15}} %V
		\put(10,-5){\line(1,0){5}}%H
		\put(15,-5){\line(0,1){5}} %V
		}
     \end{picture} 
$$}\only<5>{
\def\UNIT{1.5pt}
\begin{align*}
\setlength{\unitlength}{\UNIT}
\begin{picture}(25,20)(0,5)
	\put(6,7){$(a^c)$ }
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		\end{picture}
\otimes 
\begin{picture}(10,10)(0,2)
	%\put(2,2){\tiny $(a^p)$ }
		\multiput(0,0)(5,0){3} {\line(0,1){10}} %V
		\multiput(0,0)(0,5){3} {\line(1,0){10}} %H
		\end{picture}
	~~=~~&
\setlength{\unitlength}{\UNIT}
\begin{picture}(35,30)(0,5)
%	\put(2,2){\tiny 1 }
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){10}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){10}} %V
		\end{picture}
 ~~\oplus~~
      \setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,5)
%	\put(2,2){\tiny{2}}
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){5}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){5}} %V
		%
		\put(0,-5) {\line(1,0){5}} %H
		%
		\put(0,0) {\line(0,-1){5}} %V
		\put(5,0) {\line(0,-1){5}} %V
		\end{picture}
~~\oplus~~
\setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,5)
%	\put(2,2){\tiny{3}}
	% \thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		%
		\put(30,20) {\line(0,-1){5}} %V
		\put(35,20) {\line(0,-1){5}} %V
		%
		\put(0,-5) {\line(1,0){10}} %H
		%
		\put(0,0) {\line(0,-1){5}} %V
		\put(5,0) {\line(0,-1){5}} %V
		\put(10,0) {\line(0,-1){5}} %V
		\end{picture}
 \\
 &\oplus~~
\setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,5)
%	\put(2,2){\tiny{4}}
	% \thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){5}} %H
		\put(25,15) {\line(1,0){5}} %H
		\put(25,10) {\line(1,0){5}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		%
		\put(0,-5) {\line(1,0){5}} %H
		\put(0,-10) {\line(1,0){5}} %H
		%
		\put(0,0) {\line(0,-1){10}} %V
		\put(5,0) {\line(0,-1){10}} %V
     \end{picture}
 \oplus~~
 \setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,5)
%	\put(2,2){\tiny{5}}
	% \thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){5}} %H
		\put(25,15) {\line(1,0){5}} %H
		%
		\put(30,20) {\line(0,-1){5}} %V
		%
		\put(0,-5) {\line(1,0){10}} %H
		\put(0,-10) {\line(1,0){5}} %H
		%
		\put(0,0) {\line(0,-1){10}} %V
		\put(5,0) {\line(0,-1){10}} %V
		\put(10,0) {\line(0,-1){5}} %V
     \end{picture} 
 ~~\oplus~~
\setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,5)
%	\put(2,2){\tiny{6}}
	% \thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(0,-5) {\line(1,0){10}} %H
		\put(0,-10) {\line(1,0){10}} %H
		%
		\put(0,0) {\line(0,-1){10}} %V
		\put(5,0) {\line(0,-1){10}} %V
		\put(10,0) {\line(0,-1){10}} %V
     \end{picture}
\end{align*}
}
~\\~\\~\\~\\~\\~\\~\\~\\~\\~\\
\end{frame}


%%%%%%%%%%%%%
\begin{frame}{Exploring our new favorite tensor space, $A \otimes B \otimes C^{\otimes k}$} 
\input{Two_boundary-Brat_diagram.tex}
\end{frame}
%%%%%%%%%%%%%


\begin{frame}{} 
\alert{A two-dimensional Hecke module ($k=1$): \hfill {\small Generators: $Y_1$  and $T_0$\qquad\qquad }}
\hspace{-.6in}\begin{tabular}{cl}
\begin{minipage}{3.5cm}
\def\UNIT{.9pt}%
%\begin{tiny}
$$\xymatrix{
&
\setlength{\unitlength}{\UNIT}
\begin{picture}(25,20)(0,0)
	\put(10,22){\tiny $a$}
	\put(2,9){\tiny $c$}
	 \thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
\end{picture}		
\ar@{-}[ddl]%|{\color{alert2} 4a\!\!\phantom{|}}
\ar@{-}[dd]%|%{\color{alert2} 3a-p}
\\&\\
\setlength{\unitlength}{\UNIT}
\begin{picture}(35,30)(0,-7)
	\put(2,2){\tiny 1 }
	\thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){10}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){10}} %V
		\end{picture}
	\ar@{-}[dd]|{\color{alert2} \only<1>{-c}\only<2>{-\half(a+c)~~}}
&
\setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,-7)
	\put(2,2){\tiny{2}}
	\thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){5}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){5}} %V
		%
		\put(0,-5) {\line(1,0){5}} %H
		%
		\put(0,0) {\line(0,-1){5}} %V
		\put(5,0) {\line(0,-1){5}} %V
		\end{picture}
		\ar@{-}[ddl]|{\color{alert2} \only<1>{a}\only<2>{~\half(a+c)}\!\!\phantom{|}}
\\&\\
\setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,-10)
	\put(2,2){\tiny{5}}
	\thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){10}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){10}} %V
		%
		\put(0,-5) {\line(1,0){5}} %H
		%
		\put(0,0) {\line(0,-1){5}} %V
		\put(5,0) {\line(0,-1){5}} %V
		%
%		\put(36,16){\tiny \color{dred} 6}
%		\put(26,5){\tiny \color{dred} 14}
%		\put(6,-5){\tiny \color{dred} 15}
%		\put(0,-10){\tiny \color{dred} 16}
     \end{picture} 
&
%
}%end xy
$$\end{minipage}&
%\end{tiny}
%\columnbreak%\begin{align*}
%z_0  &= \begin{pmatrix} 4a & 0 \\ 0 & 3a -p\end{pmatrix}\\
\begin{minipage}{6cm}$${\def\TOP{1}\def\K{1}
\begin{tikzpicture}[line width=1pt , scale=.75]
\node[left] at (0,.5*\TOP) {$Y_1=$};
%%%%%%%%%%%%%
\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
\PolesTop[\TOP]
\PolesBottom[0]
\LPole[0,.5*\TOP]
\RPoleO[0,.5*\TOP]
\Over[1,0][-.5,.5*\TOP]
\Under[-.5,.5*\TOP][1,\TOP]
\LPoleO[.5*\TOP,\TOP]
\RPoleO[.5*\TOP,\TOP]
% \foreach \x in {2,...,\K}  {
%	 \draw[thin] (\x,0) -- (\x,\TOP);
%	 }
\Nodes[0][\K]
\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\node[right] at (1,.5*\TOP) {$\displaystyle = \begin{pmatrix} q^{2\only<1>{(-c)}\only<2>{(-\half(a+c))}} & 0 \\ 0 & q^{2\only<1>{(a)}\only<2>{(\half(a+c))}}\end{pmatrix}$};
\end{tikzpicture}}$$
$${\def\TOP{1}\def\K{1}
\begin{tikzpicture}[line width=1pt , scale=.75]
\node[left] at (0,.5*\TOP) {$T_0=$};
%%%%%%%%%%%%%
\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
	\PolesTop[\TOP]
	\PolesBottom[0]
	\LPoleO[0,1]
	\Under[.2,.5][1,1]
	\RPoleO[0,1]
	\Over[1,0][.2,.5]
	\Nodes[0][\K]
	\Nodes[\TOP][\K]
\node[right] at (1,.5*\TOP) {$\displaystyle \sim \begin{pmatrix}q^{2*\only<1>{(-2)}\only<2>{(-2-\half(a-c))}}&0\\0&q^{2*\only<1>{(2)}\only<2>{(2-\half(a-c))}}\end{pmatrix}$};
\end{tikzpicture}}$$
%Y_1 &= \begin{pmatrix} q^{-c} & 0 \\ 0 & q^a\end{pmatrix}\\
% &\sim \\
%\end{align*}
{\small (formulas for $T_0$ given in terms of contents of added boxes)}
\end{minipage}\end{tabular}~\\\vfill

\uncover<2->{Shift! Shift contents by $-\half(a-c+b-d) = -\half(a-c)$}

\end{frame}

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


%%%%%%%%%%%%%%%%
\begin{frame}{A concrete example of a component of $A \otimes B \otimes C \otimes C$}
\uncover<2->{\begin{tikzpicture}[scale=.5]
\node[left] at (0,-1.5) {$(a^c) = (4^3) = $};
\foreach \x in {0,...,4}{\draw (\x,0)--(\x,-3);}
\foreach \y in {0,...,3}{\draw (0,-\y)--(4,-\y);}
\end{tikzpicture}
\qquad 
\begin{tikzpicture}[scale=.5]
\node[left] at (0,-1) {$(b^d) = (2^2) = $};
\foreach \x in {0,...,2}{\draw (\x,0)--(\x,-2);}
\foreach \y in {0,...,2}{\draw (0,-\y)--(2,-\y);}
\node at (0,-2.5) {};
\end{tikzpicture}
}


$$\begin{tikzpicture}[scale=.5]
\node[left] at (0,-2.5) {$\lambda = $};
\uncover<5-6,11>{\filldraw[dgreen!30] (0,-4)--(0,-5)--(1,-5)--(1,-4);}
\uncover<7-11>{\filldraw[dgreen!30] (5,0)--(6,0)--(6,-1)--(5,-1);}
\uncover<4->{
	\filldraw[black!30] (0,0)--(0,-4)--(2,-4)--(2,-3)--(4,-3)--(4,-1)--(5,-1)--(5,0);
	}
\draw (0,0)--(7,0) (0,-1)--(7,-1) (0,-2)--(4,-2) (0,-3)--(4,-3) (0,-4)--(2,-4) (0,-5)--(1,-5);
\draw (0,0)--(0,-5) (1,0)--(1,-5) (2,0)--(2,-4) (3,0)--(3,-3) (4,0)--(4,-3) (5,0)--(5,-1) (6,0)--(6,-1) (7,0)--(7,-1);
\uncover<3->{
\draw[dgreen, line width=1.5pt] (0,0)--(4,0)--(4,-3)--(0,-3)--(0,0);
\draw[dgreen, line width=1.5pt,dotted] 
	(4,0)--(6,0)--(6,-2)--(4,-2) 
	(0,-3)--(0,-5)--(2,-5)--(2,-3);
}
\uncover<6>{\node at (5.5,-.5) {1};\node at (6.5,-.5) {2};}
\uncover<8>{\node at (.5,-4.5) {1};\node at (6.5,-.5) {2};}
\uncover<9-10>{\node at (.5,-4.5) {2};\node at (6.5,-.5) {1};}
\uncover<11->{\draw[blue]  (5,-4.5)--(0,.5) node[above left] {\tiny 0};
\node at (5.5,-.5) {\scriptsize 4.5};
\node at (.5,-4.5) {\scriptsize -4.5};
\node[right] at (0,-7) {\small Shifting by $\half (a-c+b-d) = -\half$};
}
\end{tikzpicture}$$
\uncover<4->{$\begin{tikzpicture}[scale=.25] \filldraw[black!30] (0,0)--(1,0)--(1,1)--(0,1); \draw (0,0)--(1,0)--(1,1)--(0,1)--(0,0);\end{tikzpicture} = $ boxes that must appear in the partition at level 0.}
\vfill
\uncover<10->{So the $\cH_k$-module in $A \otimes B \otimes C \otimes C$ indexed by $\lambda$ is $3$-dimensional.}
\end{frame}
%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%
\begin{frame}{} 
\input{8diml_rep_slide1b.tex}
\end{frame}
%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%
\begin{frame}{Basis of the 8-dim'l module seen as tableaux}
\def\SCALE{.2}
\vspace{-.2in}
$$\begin{tikzpicture}[scale=\SCALE]
\FRAME
\draw[thick, dotted] (0,-5)--(5,0) (10,3)--(15,8);
\only<1>{\BOX{1}{-4}{$2$}\BOX{3}{-2}{$1$}}
\only<2->{\BOX{1}{-4}{$x$}\BOX{3}{-2}{$y$}}
\dBOX{11}{4}{}\dBOX{13}{6}{}
\end{tikzpicture}
\quad 
\uncover<3->{\begin{tikzpicture}[scale=\SCALE]
\FRAME
\draw[thick, dotted] (0,-5)--(5,0) (10,3)--(15,8);
\dBOX{1}{-4}{}\BOX{3}{-2}{$y$}
\BOX{13}{6}{$x$}\dBOX{11}{4}{}
\end{tikzpicture}}
$$
\uncover<3->{
$$
\begin{tikzpicture}[scale=\SCALE]
\FRAME
\draw[thick, dotted] (0,-5)--(5,0) (10,3)--(15,8);
\dBOX{1}{-4}{}\dBOX{3}{-2}{}
\BOX{13}{6}{$x$}\BOX{11}{4}{$y$}
\end{tikzpicture}
\quad 
\begin{tikzpicture}[scale=\SCALE]
\FRAME
\draw[thick, dotted] (0,-5)--(5,0) (10,3)--(15,8);
\BOX{1}{-4}{$x$}\dBOX{3}{-2}{}
\dBOX{13}{6}{}\BOX{11}{4}{$y$}
\end{tikzpicture}
$$}
{\small\vspace{-.2in}
$$\text{where } \quad \gamma_1 = \half(a+c) - \half(b+d) \qquad \text{ and } \qquad \gamma_2 = \half(a+c) + \half (b+d),$$
\uncover<2->{and $x$ and $y$ are entries of 1 or 2. }}
\end{frame}
%%%%%%%%%%%%%%%


\begin{frame}{Back to the skew local regions picture:}
\uncover<2->{Which appear as submodules of tensor space? }
\centerline{
\begin{tikzpicture}[scale=1.875]
	\filldraw[gray!20] (0,3)--(0,0)--(3,3)--(0,3);
\draw[thick] (0,-.2)--(0,3) node[above] {$\fh^{\alpha_2}$};
\draw[thick] (-.2,-.2)--(3,3) node[above right] {$\fh^{\alpha_1}$};
\draw[gray] (-.2,0)--(3,0) node[right] {\color{gray}$\fh^{\alpha_2+2\alpha_1}$};
	\draw[dotted, thick, black!70] (3-.4,3)--(-.2,-.2+.4) node[left]{\tiny$c_2=c_1+1$};
	\draw[dotted, thick, black!70] (3,.6)--(-.2,.6) node[left] {\tiny$c_2=\gamma_1$};
	\draw[dotted, thick, black!70] (3,2.4)--(-.2,2.4) node[left] {\tiny$c_2=\gamma_2$};
	\draw[dotted, thick, black!70] (.6,-.2)--(.6,3) node[above] {\tiny$c_1=\gamma_1$};
	\draw[dotted, thick, black!70] (2.4,-.2)--(2.4,3) node[above] {\tiny$c_1=\gamma_2$};
\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]}
\uncover<1-2>{\filldraw[red!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<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);
			\filldraw[yellow!70] (.6-.15,1-.15)--(.6,1-.2)--(.6,1);
			\filldraw[red!50](.6,1-.2)--(.6+.15,1-.15)--(.6+.2,1)--(.6+.15,1+.15)--(.6,1); 
			\REG[.6,1]}
\uncover<1>{\filldraw[red!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[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+.2,2.4)--(.6+.15,2.4+.15)--(.6,2.4+.2);
		\REG[.6,2.4]}
\uncover<3>{
	\filldraw[red!50] (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);
	\REG[1.3,1.7]
	\filldraw[dgreen] (1.3,1.7) circle (1pt) node[above]{};
}
\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);
		\REG[1.3,2.4]
		\filldraw[dgreen] (1.3,2.4) circle (1pt) node[above]{};
}
\uncover<5>{
		\filldraw[red!50] (2.4,2.8)--(2.4-.15,2.8-.15)--(2.4,2.8-.2);
		\REG[2.4,2.8]
	\filldraw[dgreen] (2.4,2.8) circle (1pt) node[above]{};
}
\uncover<1>{
	% 2
	\filldraw[dgreen] (.6,2.4) circle (1pt) node[above]{};
	% 3/1
	\filldraw[dgreen] (.2,.6) circle (1pt) node[above]{};
	\filldraw[dgreen] (.6,1) circle (1pt) node[above]{};
	\filldraw[dgreen] (2,2.4) circle (1pt) node[above]{};
	\filldraw[dgreen] (2.4,2.8) circle (1pt) node[above]{};
	%4:
	\filldraw[dgreen] (.6,1.7) circle (1pt) node[above]{};
	\filldraw[dgreen] (1.3,1.7) circle (1pt) node[above]{};
	\filldraw[dgreen] (1.3,2.4) circle (1pt) node[above]{};}
	%8
\uncover<1-2>{	\filldraw[dgreen] (1,2) circle (1pt) node[above]{};}
\uncover<6->{
\filldraw[dgreen!50] (.6,1)--(.6,2.4)--(2,2.4);
\draw[line width=3pt, dgreen] (.6,1)--(.6,2.4)--(2,2.4)--(.6,1);
	\filldraw[dgreen] (.2,.6) circle (1pt) node[above]{};
	\filldraw[dgreen] (2.4,2.8) circle (1pt) node[above]{};
	\node at (1.15,2.2) {\scriptsize (lattice points)};
}
\end{tikzpicture}\hspace{-.75in}
\uncover<2-5>{\begin{tikzpicture}[scale=.19]
\FRAME
\uncover<2>{
\draw[thick, dotted] (0,-5)--(5,0) (10,3)--(15,8);
\BOX{11}{4}{$2$}\BOX{13}{6}{$1$}
\dBOX{1}{-4}{}\dBOX{3}{-2}{}
\node at (0,-15) {};}
\uncover<3>{
\draw[thick, dotted] (0,-5)--(1.5,-3) (3.5,-2)--(5,0) (10,3)--(11.5,5) (13.5,6)--(15,8);
\dBOX{1.5}{-3}{}\dBOX{2.5}{-3}{}
\BOX{11.5}{5}{$1$}\BOX{12.5}{5}{$2$}}
\uncover<4>{
\draw[thick, dotted] (0,-4)--(2,-3) (3,-2)--(5,-1) (10,4)--(12,5) (13,6)--(15,7);
\dBOX{2}{-3}{}\BOX{12}{5}{$1$}
\BOX{15}{7}{$2$}
}
\uncover<5>{\draw[thick, dotted] (0,-4)--(4,0) (11,3)--(15,7);
\BOX{15}{7}{$1$}
\BOX{16}{7}{$2$}
}
\end{tikzpicture}}
}
{$\phantom{.}\qquad$ \scriptsize(points correspond to central characters)}\\
\uncover<7->{Generalizes to a bijection between points and skew shapes, regions and tableaux.}
\end{frame}

\begin{frame}{Partitions and tableaux in general}
\uncover<2->{\begin{tikzpicture}[scale=.25]
\node[left] at (0,-1.5) {$(a^c) = (4^3) = $};
\foreach \x in {0,...,4}{\draw (\x,0)--(\x,-3);}
\foreach \y in {0,...,3}{\draw (0,-\y)--(4,-\y);}
\end{tikzpicture}
\qquad 
\begin{tikzpicture}[scale=.25]
\node[left] at (0,-1) {$(b^d) = (2^2) = $};
\foreach \x in {0,...,2}{\draw (\x,0)--(\x,-2);}
\foreach \y in {0,...,2}{\draw (0,-\y)--(2,-\y);}
%\node at (0,-2.5) {};
\end{tikzpicture}
}


$$\begin{tikzpicture}[scale=.5]
\uncover<1->{\node[left] at (0,-2.5) {$\lambda = ~$};}
\uncover<5->{
	\filldraw[blue!20] (0,-3)--(0,-5)--(1,-5)--(1,-4)--(2,-4)--(2,-3);
	\filldraw[blue!20] (4,-2)--(6,-2)--(6,0)--(5,0)--(5,-1)--(4,-1);
	}
\uncover<6->{\filldraw[yellow!50] (6,0)--(7,0)--(7,-1)--(6,-1)
							(0,-5)--(0,-6)--(1,-6)--(1,-5);}
\uncover<7->{
	\filldraw[red!20] (0,-4)--(-1,-4)--(-1,-5)--(0,-5)--(0,-4);
	\draw (0,-4)--(-1,-4)--(-1,-5)--(0,-5)--(0,-4);
	\filldraw[red!20] (5,0)--(5,1)--(6,1)--(6,0)--(5,0);
	\draw (5,0)--(5,1)--(6,1)--(6,0)--(5,0);
	}
\uncover<3>{
	\filldraw[black!30] (0,0)--(0,-3)--(4,-3)--(4,-1)--(5,-1)--(5,0);
	}
\uncover<1-3>{\draw (0,0)--(7,0) (0,-1)--(7,-1) (0,-2)--(6,-2) (0,-3)--(4,-3) (0,-4)--(2,-4) (0,-5)--(1,-5) (0,-6)--(1,-6);
\draw (0,0)--(0,-6) (1,0)--(1,-6) (2,0)--(2,-4) (3,0)--(3,-3) (4,0)--(4,-3) (5,0)--(5,-2) (6,0)--(6,-2) (7,0)--(7,-1);}
\uncover<4->{
\draw (5,0)--(7,0) (4,-1)--(7,-1) (4,-2)--(6,-2) (0,-3)--(2,-3) (0,-4)--(2,-4) (0,-5)--(1,-5) (0,-6)--(1,-6);
\draw (0,-3)--(0,-6) (1,-3)--(1,-6) (2,-3)--(2,-4) (4,-1)--(4,-2) (5,0)--(5,-2) (6,0)--(6,-2) (7,0)--(7,-1);
}
\uncover<2-3>{\draw[dgreen, line width=1pt] (0,0)--(4,0)--(4,-3)--(0,-3)--(0,0);}
\uncover<2->{\draw[dgreen, line width=1pt,dotted] 
	(4,0)--(6,0)--(6,-2)--(4,-2)--(4,0)
	(0,-3)--(0,-5)--(2,-5)--(2,-3)--(0,-3);
}
\uncover<5->{\draw[blue]  (5,-4.5)--(0,.5) node[above left] {\tiny 0};
\filldraw[blue] (3,-2.5) circle (4pt)  {};}
\uncover<9->{
	\node at (5.5,.5) {-5};
	\node at (5.5,-.5) {1};
	\node at (5.5,-1.5) {2};
	\node at (6.5,-.5) {3};
	\node at (4.5,-1.5) {-4};
	\node at (.5,-3.5) {-2};
	\node at (.5,-4.5) {-1};
	\node at (.5,-5.5) {5};
	\node at (-.5,-4.5) {-3};
	\node at (1.5,-3.5) {4};
}
\end{tikzpicture}\quad
\uncover<10>{
\begin{tikzpicture}[scale=.5]
\node[left] at (0,-2.5) {$=~$};
\filldraw[dgreen!20] (0,-3)--(0,-5)--(1,-5)--(1,-3);
\filldraw[dgreen!20] (4,-1)--(4,-2)--(5,-2)--(5,-1);
	\filldraw[black!30] (0,0)--(0,-3)--(4,-3)--(4,-1)--(5,-1)--(5,0);
\draw (0,0)--(7,0) (0,-1)--(7,-1) (0,-2)--(6,-2) (0,-3)--(4,-3) (0,-4)--(2,-4) (0,-5)--(1,-5) (0,-6)--(1,-6);
\draw (0,0)--(0,-6) (1,0)--(1,-6) (2,0)--(2,-4) (3,0)--(3,-3) (4,0)--(4,-3) (5,0)--(5,-2) (6,0)--(6,-2) (7,0)--(7,-1);
\draw[dgreen, line width=1pt] (0,0)--(4,0)--(4,-3)--(0,-3)--(0,0);
\draw[dgreen, line width=1pt,dotted] 
	(4,0)--(6,0)--(6,-2)--(4,-2)--(4,0)
	(0,-3)--(0,-5)--(2,-5)--(2,-3)--(0,-3);
\uncover<5>{\draw[blue]  (5,-4.5)--(0,.5) node[above left] {\tiny 0};}
%	\node at (5.5,.5) {-5};
	\node at (5.5,-.5) {1};
	\node at (5.5,-1.5) {2};
	\node at (6.5,-.5) {3};
%	\node at (4.5,-1.5) {-4};
%	\node at (.5,-3.5) {-2};
%	\node at (.5,-4.5) {-1};
	\node at (.5,-5.5) {5};
%	\node at (-.5,-4.5) {-3};
	\node at (1.5,-3.5) {4};
\end{tikzpicture}}
\qquad
 \uncover<4->{(k=5)}$$
\only<1-7>{\uncover<3>{$\begin{tikzpicture}[scale=.25] \filldraw[black!30] (0,0)--(1,0)--(1,1)--(0,1); \draw (0,0)--(1,0)--(1,1)--(0,1)--(0,0);\end{tikzpicture} = $ boxes that must appear in the partition at level 0.}}
\uncover<8->{\hspace{-.1in}Fill with $\{-k, \dots, -1, 1, \dots, k\}$ so that \\
\alert{1.} adjacent boxes increase down and to the right,\\
\alert{2.} rotationally symmetric boxes have opposite values,\\
\alert{3.} red boxes are negative and yellow boxes are positive. \\
The basis for the $\cH_k$-module corresponding to $\lambda$ is indexed by these tableaux.}

~


~


~

\end{frame}

%%%%%%%
\begin{frame}
  \frametitle{References}
  \footnotesize


\begin{itemize}	
	\item[[Dau\brac] Z.\ Daugherty,   {\sl Degenerate two-boundary centralizer algebras}, Pac.\ J.\ Math.,  258-1 (2012) 91--142.
%	\item [[DRV\brac] Z.\ Daugherty, A.\ Ram, R.\ Virk, {\sl Affine and graded BMW algebras I} 
	\item[[GN\brac] J.\ de Gier and A.\ Nichols,  \emph{
The two-boundary Temperley-Lieb algebra}, 
J. Algebra \textbf{321} (2009) 1132--1167.
\item[[Ram\brac] A.\ Ram, {\sl Affine Hecke algebras and generalized standard Young tableaux}, J.\ Algebra \textbf{260} (2003) 367--415.
\end{itemize}
In preparation: 
\begin{itemize} 
	\item [[DR\brac] Z.\ Daugherty, A.\ Ram, {\sl Two boundary Hecke Algebras and
the combinatorics of type $(C_n^\vee, C_n)$ Hecke algebras}  
 \end{itemize}
  

\end{frame}
%%%%%%%

\end{document}