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

\usepackage{etex} %fixes the fight that pictex has with every other drawing package
\usepackage{pictexwd}
\usepackage{tikz}
\input xy
\xyoption{all}
\xyoption{color}
\usepackage{multicol}


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


%  \definecolor{wiscbasecolor}{rgb}{1,0,.1}
%  \colorlet{wisccolor}{wiscbasecolor!70!black}
% \definecolor{alertbasecolor}{rgb}{1,.5,0}
%%  \colorlet{alert}{alertbasecolor!80!black}
%  \colorlet{alert}{wiscbasecolor!70!black}
%  \usecolortheme[named=wisccolor]{structure}

  \definecolor{basecolor}{rgb}{0,1,.4}
  \colorlet{themecolor}{basecolor!40!black}
 \definecolor{alertbasecolor}{rgb}{0,1,.4}
%  \colorlet{alert}{alertbasecolor!80!black}
  \colorlet{alert}{alertbasecolor!40!black}
  \usecolortheme[named= themecolor]{structure}


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

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


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

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

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

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


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


\def\sA{\mathscr{A}}
\def\sB{\mathscr{B}}
\def\sC{\mathscr{C}}
\def\sD{\mathscr{D}}
\def\sE{\mathscr{E}}
\def\sF{\mathscr{F}}
\def\sG{\mathscr{G}}
\def\sH{\mathscr{H}}
\def\sI{\mathscr{I}}
\def\sJ{\mathscr{J}}
\def\sK{\mathscr{K}}
\def\sL{\mathscr{L}}
\def\sM{\mathscr{M}}
\def\sN{\mathscr{N}}
\def\sO{\mathscr{O}}
\def\sP{\mathscr{P}}
\def\sQ{\mathscr{Q}}
\def\sR{\mathscr{R}}
\def\sS{\mathscr{S}}
\def\sT{\mathscr{T}}
\def\sU{\mathscr{U}}
\def\sV{\mathscr{V}}
\def\sW{\mathscr{W}}
\def\sX{\mathscr{X}}
\def\sY{\mathscr{Y}}
\def\sZ{\mathscr{Z}}



%%% Symbols %%%------------------------------------------------------------

\def\ad{\mathrm{ad}} 
\def\Aut{\mathrm{Aut}}
\def\Card{\mathrm{Card}} 
\def\diag{\mathrm{diag}} 
\def\dim{\mathrm{dim}} 
\def\ext{\mathrm{ext}} 
\def\End{\mathrm{End}} 
\def\GL{\mathrm{GL}} 
\def\Hom{\mathrm{Hom}} 
\def\half{\hbox{$\frac12$}}
\def\id{\mathrm{id}}  
\def\Ind{\mathrm{Ind}} 
\def\ind{\mathrm{ind}}  
\def\Inf{\mathrm{Inf}} 
\def\Irr{\mathrm{Irr}} 
\def\isom{\cong}
\def\one{\mathbbm{1}}
\def\O{\mathrm{O}}
\def\Rad{\mathrm{Rad}} 
\def\Res{\mathrm{Res}} 
\def\Resf{\mathrm{Resf}}
\def\SP{\mathrm{SP}}
\def\SL{\mathrm{SL}} 
\def\tr{\mathrm{tr}} 
\def\Tr{\mathrm{Tr}} 
\def\vep{\varepsilon}
\def\wt{\mathrm{wt}}

\def\d{\mathrm{d}}

\def\<{\langle}
\def\>{\rangle}
\def\varep{\varepsilon}

\def\brac{]}

\def\one{\mathbbm{1}}
\def\isom{\cong}

\def\BOX{\begin{minipage}[c]{.2cm}
\begin{tikzpicture}[scale=.2]
\draw (0,0)--(0,1) (0,0)--(1,0) (1,0)--(1,1) (0,1)--(1,1);
\end{tikzpicture}\end{minipage}}



% COLOR
\usepackage{color}
\definecolor{dred}{rgb}{.65, 0, 0.15}
\definecolor{dblue}{rgb}{0.15, 0, 0.65}
\definecolor{grey}{rgb}{.7, .7, .7}
\definecolor{dgrey}{rgb}{.1, .1, .1}
\definecolor{alert2}{rgb}{.65, 0, 0.15}




\tikzstyle over=[draw=white,double=black, double distance=1pt, line width=2pt]

\def\Over[#1,#2][#3,#4]{ %1,2=(x,y) position; 3,4=width,height
\draw[style=over]   (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);
}
\def\Under[#1,#2][#3,#4]{ %1,2=(x,y) position; 3,4=width,height
\draw  (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);
}
\def\TOP{1.5}
\def\HzTop[#1,#2]{%1=left node, 2=right node
\draw (#1,\TOP) .. controls (.5*#1+.5*#2, \TOP - .3*#2 +.3*#1) .. (#2,\TOP);}
\def\HzBot[#1,#2]{%1=left node, 2=right node
\draw (#1,0) .. controls (.5*#1+.5*#2, .2*#2 -.2*#1) .. (#2,0);}


\def\Pole[#1][#2]{%1 = bottom, 2=top
\draw (.5,#1) -- (.5,#2) (.35,#1) -- (.35,#2) ;
}
\def\PoleO[#1][#2]{%1 = bottom, 2=top
\shade[left color=white,right color=white] (.35,#1) rectangle (.5,#2);
\draw[style=over] (.5,#1) -- (.5,#1+#2) (.35,#1) -- (.35,#1+#2) ;
}
\def\PoleBottom[#1]{%1 = Vertical position
\draw (.35,#1) -- (.35,#1-.25) (.5,#1) -- (.5,#1-.25) ;
\draw (.5,#1-.25) arc (0:-180:2pt and 1.5pt);
}
\def\PoleTop[#1]{%1 = Vertical position
\draw (.35,#1) -- (.35,#1+.25) (.5,#1) -- (.5,#1+.25) ;
\draw (.5,#1+.25) arc (0:360:2pt and 1.5pt);
}
\def\PolesTop[#1]{%1 = Vertical position
\draw (.5,#1) -- (.5,#1+.25) (.35,#1) -- (.35,#1+.25) ;
\draw (.5,#1+.25) arc (0:360:2pt and 1.5pt);
\draw (-.15,#1) -- (-.15,#1+.25) (-.3,#1) -- (-.3,#1+.25) ;
\draw (-.15,#1+.25) arc (0:360:2pt and 1.5pt);
}
\def\PolesBottom[#1]{%1 = Vertical position
\draw (.5,#1) -- (.5,#1-.25) (.35,#1) -- (.35,#1-.25) ;
\draw (.5,#1-.25) arc (0:-180:2pt and 1.5pt);
\draw (-.15,#1) -- (-.15,#1-.25) (-.3,#1) -- (-.3,#1-.25) ;
\draw (-.15,#1-.25) arc (0:-180:2pt and 1.5pt);
}
\def\SymPolesTop[#1][#2]{%1 = Vertical position, 2=k
\draw (.5,#1) -- (.5,#1+.25) (.35,#1) -- (.35,#1+.25) ;
\draw (.5,#1+.25) arc (0:360:2pt and 1.5pt);
\draw (#2+.5,#1) -- (#2+.5,#1+.25) (#2+.65,#1) -- (#2+.65,#1+.25) ;
\draw (#2+.65,#1+.25) arc (0:360:2pt and 1.5pt);
}
\def\SymPolesBottom[#1][#2]{%1 = Vertical position, 2=k
\draw (.5,#1) -- (.5,#1-.25) (.35,#1) -- (.35,#1-.25) ;
\draw (.5,#1-.25) arc (0:-180:2pt and 1.5pt);
\draw (#2+.5,#1) -- (#2+.5,#1-.25) (#2+.65,#1) -- (#2+.65,#1-.25) ;
\draw (#2+.65,#1-.25)  arc (0:-180:2pt and 1.5pt);
}
\def\LPoleO[#1,#2]{%1 = bottom, 2=top
\shade[left color=white,right color=white] (-.3,#1) rectangle (-.15,#2);
\draw[style=over] (-.3,#1) -- (-.3,#2) (-.15,#1) -- (-.15,#2) ;
}
\def\RPoleO[#1,#2]{%1 = bottom, 2=top
\shade[left color=white,right color=white] (.35,#1) rectangle (.5,#2);
\draw[style=over] (.5,#1) -- (.5,#2) (.35,#1) -- (.35,#2) ;
}
\def\LPole[#1,#2]{%1 = bottom, 2=top
\draw (-.15,#1) -- (-.15,#2) (-.3,#1) -- (-.3,#2) ;
}
\def\RPole[#1,#2]{%1 = bottom, 2=top
\draw (.5,#1) -- (.5,#2) (.35,#1) -- (.35,#2) ;
}
\def\RRPole[#1,#2][#3]{%1 = bottom, 2=top, 3=k
\draw (#3+.5,#1) -- (#3+.5,#2) (#3+.65,#1) -- (#3+.65,#2) ;
}
\def\RRPoleO[#1,#2][#3]{%1 = bottom, 2=top
\shade[left color=white,right color=white] (#3+.65,#1) rectangle (#3+.5,#2);
\draw[style=over] (#3+.5,#1) -- (#3+.5,#2) (#3+.65,#1) -- (#3+.65,#2) ;
}
\def\PoleTwist[#1,#2]{%1 = bottom, 2=top
\draw (.15,#1) .. controls ++(0,#2*.5-#1*.5) and ++(0,-#2*.5+#1*.5) ..  (.5,#2) 
			(0,#1)   .. controls ++(0,#2*.55-#1*.55) and ++(0,-#2*.45+#1*.45) ..    (.35,#2) ;
\draw[thin,over] (.5,#1) .. controls ++(0,#2*.5-#1*.5) and ++(0,-#2*.5+#1*.5) ..  (.15,#2) 
		(.35,#1)  .. controls ++(0,#2*.45-#1*.45) and ++(0,-#2*.55+#1*.55) ..   (0,#2) ;
}
\def\Nodes[#1][#2]{
 \foreach \x in {1,...,#2} {
%       \draw (\x,0) circle (0.4cm);
	 \filldraw [black] (\x,#1) circle (2pt);
	 }
	}
\def\Tops[#1][#2]{
 \foreach \x in {1,...,#2} {
	%       \draw (\x,0) circle (0.4cm);
	 \filldraw [black] (\x,#1+.4) circle (2.5pt);
	 }
	 \foreach \x in {1,...,#2} {
	%       \draw (\x,0) circle (0.4cm);
		 \draw (\x,#1) -- (\x,#1+.4) ;
	 }
}
\def\Bottoms[#1][#2]{
 \foreach \x in {1,...,#2} {
%       \draw (\x,0) circle (0.4cm);
	 \filldraw [black] (\x,#1-.4) circle (2.5pt);
	 }
	 \foreach \x in {1,...,#2} {
%       \draw (\x,0) circle (0.4cm);
	 \draw (\x,#1) -- (\x,#1-.4) ;
	 }
	}
\def\Label[#1,#2][#3][#4]{
	\draw (#3,#2+1) node{#4};
	\draw (#3,#1-1) node{#4};
}




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

\title[Type C symmetry of two-boundary Hecke algebras]{Type C symmetry of \\ two-boundary Hecke algebras}
\author[Zajj Daugherty]{Zajj Daugherty\\~\\
{\small Joint with Arun Ram}}
\institute{Dartmouth College}
\date{March 14, 2012\\
(Happy $\pi$ day!)}
\begin{document}

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



\section{A quick tour of some diagram algebras}
%%%%%%%%
\begin{frame}{A quick tour of some diagram algebras}
Everyone's favorite diagram algebra: \\
Group algebra of the {\bf symmetric group} $S_k$

$$\begin{tikzpicture}[line width=1pt]
\uncover<3->{
	\foreach \x in {1,...,5}{
	\draw[dotted, themecolor] (\x,0)--(\x,-.5);
	}
}
%%%%%%%%%%%%%
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
\node at (\x, 1.8) {\scriptsize \color{themecolor} \x};
\uncover<1-2>{  \node at (\x, -.3) {\scriptsize \color{themecolor} \x};}
}
\draw[->, shorten >=2pt] (1,1.5)--(2,0); 
\draw[->, shorten >=2pt] (2,1.5)--(4,0); 
\draw[->, shorten >=2pt] (3,1.5)--(1,0); 
\draw[->, shorten >=2pt] (4,1.5)--(5,0); 
\draw[->, shorten >=2pt] (5,1.5)--(3,0); 
%%%%%%%%%%%%%
\pgftransformyshift{-2.1cm}
%%%%%%%%%%%%%
\uncover<2->{
	\foreach \x in {1,...,5}{
		 \filldraw [black] (\x,1.5) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
		  \node at (\x, -.3) {\scriptsize \color{themecolor} \x};
	 }
	\draw[->, shorten >=2pt] (1,1.5)--(2,0); 
	\draw[->, shorten >=2pt] (2,1.5)--(1,0); 
	\draw[->, shorten >=2pt] (3,1.5)--(4,0); 
	\draw[->, shorten >=2pt] (4,1.5)--(5,0); 
	\draw[->, shorten >=2pt] (5,1.5)--(3,0); 
}
%%%%%%%%%%%%%
\pgftransformyshift{1cm}
\pgftransformxshift{6cm}
%%%%%%%%%%%%%
\uncover<4->{\node at (0,.75) {$=$};
	\foreach \x in {1,...,5}{
		 \filldraw [black] (\x,1.5) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
		  \node at (\x, -.3) {\scriptsize \color{themecolor} \x};
		  \node at (\x, 1.8) {\scriptsize \color{themecolor} \x};
	 }
	\draw[->, shorten >=2pt] (1,1.5)--(1,0); 
	\draw[->, shorten >=2pt] (2,1.5)--(5,0); 
	\draw[->, shorten >=2pt] (3,1.5)--(2,0); 
	\draw[->, shorten >=2pt] (4,1.5)--(3,0); 
	\draw[->, shorten >=2pt] (5,1.5)--(4,0); 
}
\end{tikzpicture}$$
\uncover<2->{(with multiplication given by concatenation)}
\end{frame}



%%%%%%%%
\begin{frame}{A quick tour of some diagram algebras}
More examples: \\
Group algebra of the {\bf braid group}

$$\begin{tikzpicture}[line width=1pt]
\node at (1, 1.8) {\scriptsize \color{white} 1};
\node at (1, -.3) {\scriptsize \color{white} 1};
\node at (1, -2.4) {\scriptsize \color{white} 1};
%%%%%%%%%%%%%
	\foreach \x in {1,...,5}{
	\draw[dotted, themecolor] (\x,0)--(\x,-.5);
	}
%%%%%%%%%%%%%
\draw[style=over]  (1,1.5)--(2,0); 
\draw[style=over]  (2,1.5)--(4,0); 
\draw[style=over]  (3,1.5)--(1,0); 
\draw[style=over]  (4,1.5)--(5,0); 
\draw[style=over]   (5,1.5)--(3,0); 
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);}
%%%%%%%%%%%%%
\pgftransformyshift{-2.1cm}
%%%%%%%%%%%%%
	\draw[style=over] (1,1.5)--(2,0); 
	\draw[style=over]  (2,1.5)--(1,0); 
	\draw[style=over]  (3,1.5)--(4,0); 
	\draw[style=over]  (4,1.5)--(5,0); 
	\draw[style=over]  (5,1.5)--(3,0); 
	\foreach \x in {1,...,5}{
		 \filldraw [black] (\x,1.5) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
	 }
%%%%%%%%%%%%%
\pgftransformyshift{1cm}
\pgftransformxshift{6cm}
%%%%%%%%%%%%%
\uncover<2->{\node at (0,.75) {$=$};
	\draw (2,1.5).. controls (4,1) .. (5,0);
	\Under[1,1.5][1.75,.75];
	\Under[1.25,.75][2,0];
	\Under[4,1.5][4.25,.75];
	\Over[3,1.5][1.25,.75];
	\Over[5,1.5][4,.75];
	\Over[1.75,.75][1,0];
	\Over[4,.75][4,0];
	\Over[4.25,.75][3,0];
	\foreach \x in {1,...,5}{
		 \filldraw [black] (\x,1.5) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
		  \node at (\x, -.3) {\scriptsize \color{themecolor} \x};
		  \node at (\x, 1.8) {\scriptsize \color{themecolor} \x};
	 }
}
\end{tikzpicture}$$
(with multiplication given by concatenation)
\end{frame}




%%%%%%%%
\begin{frame}{A quick tour of some diagram algebras}

\begin{tabular}{cc}
{\bf Brauer algebras}  & {\bf Temperley-Lieb algebras} \\ 
$\begin{tikzpicture}[line width=1pt]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
\HzTop[1,2];
 \HzBot[2,4];
\draw (3,1.5)--(1,0); 
\draw (4,1.5)--(5,0); 
\draw (5,1.5)--(3,0); 
\end{tikzpicture}$
&
$\begin{tikzpicture}[line width=1pt]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
\HzTop[1,2];
 \HzBot[3,4];
\draw (3,1.5)--(1,0); 
\draw (4,1.5)--(2,0); 
\draw (5,1.5)--(5,0); 
\end{tikzpicture}$
\\
\multicolumn{2}{c}{\small (with relations on closed loops)}
\\~\\
{\bf BMW algebras} & {\bf Affine braid group} \\
\begin{minipage}[c]{2in}
$\begin{tikzpicture}[line width=1pt]
\HzTop[1,2];
 \HzBot[2,4];
\draw (3,1.5)--(1,0); 
\draw[over] (4,1.5)--(5,0); 
\draw[over] (5,1.5)--(3,0); 
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
\end{tikzpicture}$\end{minipage}
&
\begin{minipage}[c]{2in}$\begin{tikzpicture}[line width=1pt]
\PoleTop[\TOP]
\PoleBottom[0]
	\draw (2,1.5).. controls (4,1) .. (5,0);
	\Under[1,1.5][1.75,.75];
	\Under[.25,.75][2,0];
	\PoleO[0][.5*\TOP]
	\Under[4,1.5][4.25,.75];
	\Pole[.5*\TOP][\TOP]
	\Over[3,1.5][.25,.75];
	\Over[5,1.5][4,.75];
	\Over[1.75,.75][1,0];
	\Over[4,.75][4,0];
	\Over[4.25,.75][3,0];
	\foreach \x in {1,...,5}{
		 \filldraw [black] (\x,1.5) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
	 }
\end{tikzpicture}$\end{minipage}\\
\begin{minipage}{2.1in}
\begin{center}\small (With relations on closed loops and crossings, a la Skein relations)\end{center}\end{minipage}
\end{tabular}

~

(Affine) {\bf Hecke algebras} of type A are quotients of the (affine) braid group by relations on double twists.
\end{frame}



\section{Actions on tensor space}
\begin{frame}{Actions on tensor space}
{} {\bf Classical example:} (Schur 1901) \\
\begin{enumerate} 
\item<2->  $\GL_n(\CC)$ acts on $\CC^n \otimes \CC^n \otimes \cdots \otimes \CC^n = \left( \CC^n\right)^{\otimes k}$
diagonally.
$$g \cdot (v_1 \otimes v_2 \otimes \cdots \otimes v_k) = g v_1 \otimes g v_2 \otimes \cdots \otimes g v_k.$$
\item<3-> $S_k$ also acts on  $\left( \CC^n\right)^{\otimes k}$ by place permutation.
$$\begin{tikzpicture}[line width=1pt, scale=.9]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
\node at (\x, 1.3) {\small \color{themecolor} $v_{\x}$};
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \color{themecolor} $\otimes$};
\node at (\x+.5, -.3) {\small \color{themecolor} $\otimes$};
}
\node at (1, -.3) {\small \color{themecolor}$v_{3}$};
\node at (2, -.3) {\small \color{themecolor} $v_{1}$};
\node at (3, -.3) {\small \color{themecolor} $v_{5}$};
\node at (4, -.3) {\small \color{themecolor} $v_{2}$};
\node at (5, -.3) {\small \color{themecolor} $v_{4}$};
\draw[->, shorten >=2pt] (1,1)--(2,0); 
\draw[->, shorten >=2pt] (2,1)--(4,0); 
\draw[->, shorten >=2pt] (3,1)--(1,0); 
\draw[->, shorten >=2pt] (4,1)--(5,0); 
\draw[->, shorten >=2pt] (5,1)--(3,0); 
\end{tikzpicture}$$
\item<4-> These actions commute!  
$$\begin{minipage}{4.5cm}\begin{tikzpicture}[line width=1pt, scale=.9]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
\node at (\x, 1.3) {\small \color{blue}$g$\color{themecolor}$v_{\x}$};
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \color{themecolor} $\otimes$};
\node at (\x+.5, -.3) {\small \color{themecolor} $\otimes$};
}
\node at (1, -.3) {\small \color{blue}$g$\color{themecolor}$v_{3}$};
\node at (2, -.3) {\small \color{blue}$g$\color{themecolor}$v_{1}$};
\node at (3, -.3) {\small \color{blue}$g$\color{themecolor}$v_{5}$};
\node at (4, -.3) {\small \color{blue}$g$\color{themecolor}$v_{2}$};
\node at (5, -.3) {\small \color{blue}$g$\color{themecolor}$v_{4}$};
\draw[->, shorten >=2pt] (1,1)--(2,0); 
\draw[->, shorten >=2pt] (2,1)--(4,0); 
\draw[->, shorten >=2pt] (3,1)--(1,0); 
\draw[->, shorten >=2pt] (4,1)--(5,0); 
\draw[->, shorten >=2pt] (5,1)--(3,0); 
\end{tikzpicture}\end{minipage} \text{ vs. } 
\begin{minipage}{4.5cm}\begin{tikzpicture}[line width=1pt, scale=.9]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
\node at (\x, 1.3) {\small \color{themecolor}$v_{\x}$};
}
\foreach \x in {1,...,4}{
\node at (\x+.5, 1.3) {\small \color{themecolor} $\otimes$};
\node at (\x+.5, -.3) {\small \color{themecolor} $\otimes$};
}
\node at (1, -.3) {\small \color{blue}$g$\color{themecolor}$v_{3}$};
\node at (2, -.3) {\small \color{blue}$g$\color{themecolor}$v_{1}$};
\node at (3, -.3) {\small \color{blue}$g$\color{themecolor}$v_{5}$};
\node at (4, -.3) {\small \color{blue}$g$\color{themecolor}$v_{2}$};
\node at (5, -.3) {\small \color{blue}$g$\color{themecolor}$v_{4}$};
\draw[->, shorten >=2pt] (1,1)--(2,0); 
\draw[->, shorten >=2pt] (2,1)--(4,0); 
\draw[->, shorten >=2pt] (3,1)--(1,0); 
\draw[->, shorten >=2pt] (4,1)--(5,0); 
\draw[->, shorten >=2pt] (5,1)--(3,0); 
\end{tikzpicture}\end{minipage}$$

\end{enumerate}
\end{frame}

\begin{frame}{Schur-Weyl duality}
{} {\bf Classical example:} $S_k$ and $\GL_n$ have commuting actions on $\left( \CC^n\right)^{\otimes k}$.
 Even better, if $k\leq n$,
$$\End_{\CC \GL_n}\left(\left( \CC^n\right)^{\otimes k} \right) = \CC S_k \quad \text{ and } \quad \End_{\CC S_k}\left( \left( \CC^n\right)^{\otimes k}\right) = \CC \GL_n. $$
\uncover<2->{\begin{block}{\bf Why this is exciting:}
		Centralizer relationship produces
			$$\left( \CC^n\right)^{\otimes k}  \cong \bigoplus_{\tiny \lambda \vdash k }  G^\lambda \otimes S^\lambda
			\quad \text{ as a $\GL_n$-$S_k$ bimodule,}$$
where 		\begin{tabular}{rcl}
			$G^\lambda$ &are distinct irreducible& $\GL_n$-modules\\
			$S^\lambda$ &are distinct irreducible& $S_k$-modules\\
			\end{tabular}
	\end{block}}
	
	\uncover<3->{{\bf Punchline:} Knowing a lot about symmetric group modules now produces information about  $\GL_n$-modules. }
\end{frame}



\begin{frame}{Diagram algebras as centralizer algebras}
\small
\begin{enumerate}
\item<1-> The Brauer algebras centralize the actions of $\O_n$ and $\mathrm{SP}_n$ ($n$ even) on $(\CC^n)^{\otimes k}$. (Brauer 1937)
\item<2-> The group algebra of the \only<5->{(\alert{affine}) }braid group commutes with the quantum group $U_q\fg$ on $\only<5->{\alert{M\otimes}} V^{\otimes k}$, and has centralizers as quotients.\\ \uncover<3->{If $V = L(\BOX)$  \hfill  \uncover<5->{{\small (See Orellana-Ram 2007)}} \\~\\
\centerline{\uncover<5->{(\alert{affine})} Temperley-Lieb algebras arise if $\fg = \fgl_2$ or $\fsl_2$}
\centerline{\uncover<5->{(\alert{affine})} Hecke algebras arise if $\fg = \fgl_n$ or $\fsl_n$}
\centerline{\uncover<5->{(\alert{affine})} BMW algebras arise if $\fg = \fso_n$ or $\fsp_{2n}$} }
\end{enumerate}
\uncover<4->{Idea: the picture encodes a map from $\only<5->{\alert{M\otimes}} V \otimes \cdots \otimes V$ to itself.
$$\begin{tikzpicture}[line width=1pt]
\uncover<5->{\PoleTop[\TOP]
\PoleBottom[0]
\Pole[0][\TOP]}
\HzTop[1,2];
 \HzBot[2,4];
\draw (3,1.5)--(1,0); 
\draw[over] (4,1.5)--(5,0); 
\draw[over] (5,1.5)--(3,0); 
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,1.5) circle (2pt);
 \filldraw [black] (\x, 0) circle (2pt);
 \node at (\x, 2) {\scriptsize $V$};
  \node at (\x, -.5) {\scriptsize $V$};
}
\foreach \x in {1,...,4}{
 \node at (\x+.5, 2) {\scriptsize $\otimes$};
  \node at (\x+.5, -.5) {\scriptsize $\otimes$};
}
\uncover<5->{ \node at (.415, 2) {\scriptsize $M$};
  \node at (.415, -.5) {\scriptsize $M$};
 \node at (.725, 2) {\scriptsize $\otimes$};
  \node at (.725, -.5) {\scriptsize $\otimes$};}
\end{tikzpicture}$$}
 
\end{frame}



\section{Two-boundary algebras}
\begin{frame}{Two-boundary algebras}
J.\ de Gier, A.\ Nicols, et.\ al.\ (2009):\\
 Two-boundary Temperley-Lieb algebra $\cT_k$
{\def\K{5}
$$\begin{tikzpicture}[line width=1pt ]
\SymPolesTop[\TOP][\K]
\SymPolesBottom[0][\K]
\HzTop[1,2];
 \HzBot[3,4];
\Under[3,\TOP][.15, .3*\TOP]
\RPoleO[0,\TOP]
\Over[.15, .3*\TOP][1,0]
\draw (4,\TOP)--(2,0); 
\Under[5,\TOP][5.85,.5*\TOP] 
\RRPoleO[0,\TOP][\K]
\Over[5.85,.5*\TOP][5,0]
\foreach \x in {1,...,5}{
 \filldraw [black] (\x,\TOP) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
\uncover<2->{
  \node at (.415, 2) {\scriptsize $N$};
  \node at (.415, -.5) {\scriptsize $N$};
    \node at (5.565, 2) {\scriptsize $M$};
  \node at (5.565, -.5) {\scriptsize $M$};
  \node at (.715, 2) {\scriptsize $\otimes$};
  \node at (.715, -.5) {\scriptsize $\otimes$};
  \node at (5.3, 2) {\scriptsize $\otimes$};
  \node at (5.3, -.5) {\scriptsize $\otimes$};
  \foreach \x in {1,...,5}{
 	\node at (\x, 2) {\scriptsize $V$};
 	\node at (\x, -.5) {\scriptsize $V$};
  }
  \foreach \x in {1,...,4}{
 	\node at (\x+.5, 2) {\scriptsize $\otimes$};
 	\node at (\x+.5, -.5) {\scriptsize $\otimes$};
  }
}
\end{tikzpicture}$$}%
\begin{enumerate}
\item<2-> $\cT_k$ acts on $N \otimes V^{\otimes k} \otimes M$, where $N$, $M$, and $V = L(\BOX)$ are $U_q \fsl_2$-modules (small type A). 
\item<3->  $\cT_k$ is a quotient of the affine Hecke algebra of type C (a new  and exciting character in our story). 
\end{enumerate}


\uncover<4->{{\bf Question:} Can we lift the pictures and commutator results up to the Hecke algebra by studying tensor products of $U_q \fsl_n$-modules?}

\end{frame}


\begin{frame}{Type C affine Hecke algebra}

$$\begin{tikzpicture}[scale=1.2]
\uncover<3->{
	\draw[thick,blue, double, double distance=2pt] (6,0)--(5,0);
	\filldraw[white] (6,0) circle (4pt);
	\draw[thick,blue] (6,0) circle (4pt);
	\node at (6,.4) {\footnotesize {\color{blue} $k$}};
}
\foreach \x in {1,2, 4} {
	\draw[thick] (\x,0) -- (\x + 1, 0);
	}
\node at (3.5,0) {$\cdots$};
\foreach \x in {0,1,2, 3} {
	\node at (\x,.4) {\footnotesize \alert{\x}};
	}
\node at (4,.4) {\footnotesize \alert{$k\!-\!2$}};
\node at (5,.4) {\footnotesize \alert{$k\!-\!1$}};
\draw[thick, double, double distance=2pt] (0,0)--(1,0);
\foreach \x in {0, ..., 5} {
	\filldraw[white] (\x,0) circle (4pt);
	\draw[thick] (\x,0) circle (4pt);
	}
\end{tikzpicture}$$


\uncover<2->{The {\bf Weyl group of type C} is generated by  $s_0, \dots, s_{k-1}$ with relations  $s_i^2 = 1$ and }
$$\uncover<2->{\underbrace{s_i s_j \dots }_{m_{i,j} \text{ factors}} =  \underbrace{s_j s_i \dots }_{m_{i,j} \text{ factors}}
\qquad  \text{ where }}
 \qquad  m_{i,j} =  \quad \begin{matrix}
\text{\small$2$ \quad  if } \quad 
	 	\begin{minipage}[t]{1cm}
		\begin{tikzpicture}[scale=.75]
			\filldraw[white] (1,0) circle (4pt);
			\draw[thick] (1,0) circle (4pt);
			\node at (1,.4) {\footnotesize \alert{$i$}};
			\filldraw[white] (2,0) circle (4pt);
			\draw[thick] (2,0) circle (4pt);
			\node at (2,.4) {\footnotesize \alert{$j$}};
		\end{tikzpicture}\end{minipage}\\ 
\text{\small$3$ \quad  if } \quad 
	 	\begin{minipage}[t]{1cm}
		\begin{tikzpicture}[scale=.75]
			\draw[thick] (1,0)--(2,0);
			\filldraw[white] (1,0) circle (4pt);
			\draw[thick] (1,0) circle (4pt);
			\node at (1,.4) {\footnotesize \alert{$i$}};
			\filldraw[white] (2,0) circle (4pt);
			\draw[thick] (2,0) circle (4pt);
			\node at (2,.4) {\footnotesize \alert{$j$}};
		\end{tikzpicture}\end{minipage}\\
\text{\small$4$ \quad  if } \quad 
	 	\begin{minipage}[t]{1cm}
		\begin{tikzpicture}[scale=.75]
		\draw[thick, double, double distance=2pt] (1,0)--(2,0);
			\filldraw[white] (1,0) circle (4pt);
			\draw[thick] (1,0) circle (4pt);
			\node at (1,.4) {\footnotesize \alert{$i$}};
			\filldraw[white] (2,0) circle (4pt);
			\draw[thick] (2,0) circle (4pt);
			\node at (2,.4) {\footnotesize \alert{$j$}};
		\end{tikzpicture}\end{minipage}\end{matrix}
$$%
\uncover<3->{Fix  constants $a_0, a_{k},$ and $a_1=\cdots=a_{k-1}$. The {\bf affine Hecke algebra of type C} is generated by $T_0, T_1, \dots, T_{k}$ with relations 
$$T_i^2 = (a_i - a_i^{-1}) T_i + 1, \qquad \underbrace{T_i T_j \dots }_{m_{i,j} \text{ factors}} =  \underbrace{T_j T_i \dots }_{m_{i,j} \text{ factors}}.$$}
\end{frame}

\begin{frame}{Why the two-boundary braid group is type C}
The {\bf two-boundary (two-pole) braid group} is generated by 
\centerline{\def\TOP{1}\def\K{1}
\begin{tikzpicture}[line width=1pt , scale=.75]
\node at (0,.5*\TOP) {$T_k= \cdots$};
%%%%%%%%%%%%%%
\pgftransformxshift{0cm}
%%%%%%%%%%%%%%
\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,0][\K+.8,.5]
\RRPoleO[0,1][\K]
\Over[\K+.8,.5][\K,1]
\Nodes[0][\K]
\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{4 cm}
%%%%%%%%%%%%%
\node[left] at (0,.5*\TOP) {$T_0=$};
%%%%%%%%%%%%%
%\pgftransformxshift{.5cm}
%%%%%%%%%%%%%
	\PoleTop[\TOP]
	\PoleBottom[0]
	\Under[.2,.5][1,1]
	\RPoleO[0,1]
	\Over[1,0][.2,.5]
	\Nodes[0][\K]
	\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{4 cm}
%%%%%%%%%%%%%
\node[left] at (1,.5*\TOP) {$\cdots \quad$ and \quad $T_i=$};
%%%%%%%%%%%%%%
%\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%%
	\Under[1,0][2,\TOP]
	\Over[2,0][1,\TOP]
	\Nodes[0][2]
	\Nodes[\TOP][2]
	\Label[.5,.5][1][\footnotesize $i$]
	\Label[.5,.5][2][\footnotesize$i$+1]
%%%%%%%%%%%%%
\pgftransformxshift{2 cm}
\pgftransformxshift{.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {\qquad for $1\leq i \leq k-1$.};
\end{tikzpicture}
}


 

\uncover<2->{\centerline{${\begin{tikzpicture}[scale=1.2]
\foreach \x in {1,2,3,5} {
	\draw[thick] (\x,0) -- (\x + 1, 0);
	}
\node at (4.5,0) {$\cdots$};
\draw[thick, double, double distance=2pt] (6,0)--(7,0);
\foreach \x in {1, ..., 7} {
	\filldraw[white] (\x,0) circle (4pt);
	\draw[thick] (\x,0) circle (4pt);
	}
\only<4>{\draw[thick,blue, double, double distance=2pt] (0,0)--(1,0);
	\filldraw[white] (0,0) circle (4pt);
	\filldraw[white] (1,0) circle (4pt);
	\draw[thick,blue] (0,0) circle (4pt);
	\draw[thick,blue] (1,0) circle (4pt);
}
\only<1,2,3>{\draw[thick, double, double distance=2pt] (0,0)--(1,0);
	\filldraw[white] (0,0) circle (4pt);
	\filldraw[white] (1,0) circle (4pt);
	\draw[thick](1,0) circle (4pt);
	\draw[thick] (0,0) circle (4pt);
}
\only<3>{
	\draw[thick, blue] (2,0)--(3,0);
	\filldraw[white] (2,0) circle (4pt);
	\filldraw[white] (3,0) circle (4pt);
	\draw[thick,blue](3,0) circle (4pt);
	\draw[thick,blue] (2,0) circle (4pt);
}
\end{tikzpicture}}$}}%
\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}{Why the two-boundary braid group is type C}
\begin{thm}[\small D.-Ram, degenerate version in [Da\brac]{\small Let $U=U_q \fg$ for any complex reductive Lie algebras $\fg$. Let $N$, $M$, and $V$ be finite-dimensional modules. }

 The two-boundary braid group acts on 
$$N \otimes (V)^{\otimes k} \otimes M$$
and this action commutes with the action of $U$. 
\end{thm}

\vfill

{\small How?\\
Quantum group setting: The action is produced via $R$-matrices.\\
Lie algebra setting: The action is produced via the Casimir.}
\end{frame}



\begin{frame}{Why the representation theory is type A} 

Let $U = U_q \fsl_n$. 


\uncover<2->{The finite-dimensional $U$-modules $L(\lambda)$ are indexed by \emph{partitions}:
$$\lambda = \begin{picture}(40, 23)(0,2)
		\multiput(0,0)(10,0){4} {\line(0,1){20}} %V
		\multiput(0,0)(0,10){3} {\line(1,0){30}} %H
		%
		\put(30,20) {\line(1,0){10}} %H
		\put(30,10) {\line(1,0){10}} %H
		%
		\put(40,20) {\line(0,-1){10}} %V
		%
		\put(0,-10) {\line(1,0){10}} %H
		%
		\put(0,0) {\line(0,-1){10}} %V
		\put(10,0) {\line(0,-1){10}} %V
		%
		\only<3->{
		\put(3,12){\text{\footnotesize\alert{0}}}
		\put(1,2){\text{\footnotesize\alert{-1}}}
		\put(1,-8){\text{\footnotesize\alert{-2}}}
		%
		\put(13,12){\text{\footnotesize\alert{1}}}
		\put(13,2){\text{\footnotesize\alert{0}}}
		%
		\put(23,12){\text{\footnotesize\alert{2}}}
		\put(23,2){\text{\footnotesize\alert{1}}}
		%
		\put(33,12){\text{\footnotesize\alert{3}}}
		}
		\only<2>{
		\put(50,12){\small \color{alert2} 4}
		\put(50,2){\small \color{alert2} ~+3}
		\put(50,-8){\small \color{alert2} ~+1}
		}
     \end{picture} 
$$
~
\centerline{\small (a collection of boxes piled up and to the left)}}
\uncover<3->{%
If a box $B$ is in row $i$ and column $j$, then the \emph{content} of $B$ is \\
	\centerline{$c(B) = j - i.$}}%

~

\uncover<4->{%
If $M = L(\mu)$, $N = L(\nu)$, and $V=L(\BOX)$,\\
\centerline{eigenvalues($T_0$) $\sim$ $\left\{q^{c(\text{addable boxes of $\nu$})}\right\}$}
\centerline{eigenvalues($T_i$) $\sim$ $\left\{q, q^{-1}\right\}$ for $1 \leq i \leq k-1$}
\centerline{eigenvalues($T_k$) $\sim$ $\left\{q^{c(\text{addable boxes of $\mu$})}\right\}$}}


~

\uncover<5->{%
Back to the Hecke algebra: The relations $T_i^2 = (a_i - a_i^{-1}) T_i + 1$  say that $T_i$ has {\bf two} eigenvalues. }

\end{frame}


\begin{frame}{Why the representation theory is type A} 
Back to the Hecke algebra: The relations $T_i^2 = (a_i - a_i^{-1}) T_i + 1$  say that $T_i$ has {\bf two} eigenvalues.

\smallskip 

\uncover<2->{%
So $\mu$ and $\nu$ must be rectangles!\\
(Exactly two addable boxes)
	$$
	(a^c) = \begin{picture}(60, 18)(-10,12)
		\multiput(10,2)(10,0){3} {\line(0,1){26}} %V
		\multiput(2,10)(0,10){2} {\line(1,0){36}} %H
		%
		\thicklines
		\alert{
		\put(0,-10) {\line(1,0){10}} %H
		%
		\put(0,0) {\line(0,-1){10}} %V
		\put(10,0) {\line(0,-1){10}} %V
		%
		\put(40,30) {\line(1,0){10}} %H
		\put(40,20) {\line(1,0){10}} %H
		%
		\put(50,30) {\line(0,-1){10}} %V
		}		
		\put(1,-7){\text{\footnotesize\alert{-$c$}}}
		\put(43,23){\text{\footnotesize\alert{$a$}}}
		\multiput(0,0)(40,0){2} {\line(0,1){30}} %V
		\multiput(0,0)(0,30){2} {\line(1,0){40}} %H
		%
		\put(-10,12){$c$}
		\put(17,32){$a$}
		%
     \end{picture} 
     $$
}


\uncover<3->{\begin{thm}[\small D.-Ram in progress, degenerate version in [Da\brac]{\small Let $U=U_q \fsl_n$ and $a_1 = \dots = a_{k-1} = q$.}

 The Hecke algebra of type C acts on 
$$L((b^d)) \otimes (L(\BOX))^{\otimes k} \otimes L((a^c))$$
and this action commutes with the action of $U$. 
\end{thm}}


\end{frame}

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

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


\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}{Why the representation theory is type A} 
~\\
Let $M = L((a^c))$ and $N = L((b^d))$. 
Then 
	$$M \otimes N = \bigoplus_{\lambda \in \Lambda} L(\lambda) \qquad \text{\tiny (multiplicity one!)}$$

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}{Why the representation theory is type A} 
\input{Two_boundary-Brat_diagram.tex}
\end{frame}
%%%%%%%%%%%%%


\begin{frame}{Why the representation theory is type A} 
\alert{A two-dimensional Hecke module ($k=1$): \hfill {\small Generators: $ Y_1$  and $T_0$\qquad\qquad }}
\begin{multicols}{2}
\def\UNIT{.9pt}%
%\begin{tiny}
$$\xymatrix{
&
\setlength{\unitlength}{\UNIT}
\begin{picture}(25,20)(0,0)
	\put(10,22){\tiny $a$}
	\put(2,9){\tiny $c$}
	 \thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
\end{picture}		
\ar@{-}[ddl]%|{\color{alert2} 4a\!\!\phantom{|}}
\ar@{-}[dd]%|%{\color{alert2} 3a-p}
\\&\\
\setlength{\unitlength}{\UNIT}
\begin{picture}(35,30)(0,-7)
	\put(2,2){\tiny 1 }
	\thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){10}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){10}} %V
		\end{picture}
	\ar@{-}[dd]|{\color{alert2} -c}
&
\setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,-7)
	\put(2,2){\tiny{2}}
	\thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){5}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){5}} %V
		%
		\put(0,-5) {\line(1,0){5}} %H
		%
		\put(0,0) {\line(0,-1){5}} %V
		\put(5,0) {\line(0,-1){5}} %V
		\end{picture}
		\ar@{-}[ddl]|{\color{alert2} a\!\!\phantom{|}}
\\&\\
\setlength{\unitlength}{\UNIT}
\begin{picture}(35, 30)(0,-10)
	\put(2,2){\tiny{5}}
	\thicklines
		\multiput(0,0)(25,0){2} {\line(0,1){20}} %V
		\multiput(0,0)(0,20){2} {\line(1,0){25}} %H
		%
		\put(25,20) {\line(1,0){10}} %H
		\put(25,15) {\line(1,0){10}} %H
		\put(25,10) {\line(1,0){10}} %H
		%
		\put(30,20) {\line(0,-1){10}} %V
		\put(35,20) {\line(0,-1){10}} %V
		%
		\put(0,-5) {\line(1,0){5}} %H
		%
		\put(0,0) {\line(0,-1){5}} %V
		\put(5,0) {\line(0,-1){5}} %V
		%
%		\put(36,16){\tiny \color{dred} 6}
%		\put(26,5){\tiny \color{dred} 14}
%		\put(6,-5){\tiny \color{dred} 15}
%		\put(0,-10){\tiny \color{dred} 16}
     \end{picture} 
&
%
}%end xy
$$
%\end{tiny}
\columnbreak\begin{align*}
%z_0  &= \begin{pmatrix} 4a & 0 \\ 0 & 3a -p\end{pmatrix}\\
Y_1 &= \begin{pmatrix} q^{-c} & 0 \\ 0 & q^a\end{pmatrix}\\
T_0 &\sim \begin{pmatrix}q^{-2}&0\\0&q^2\end{pmatrix}\\
\end{align*}{\small (formulas for $T_0$ given in terms of contents of added boxes)}


\end{multicols}
\end{frame}

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





%%%%%%%%%%%%%%%%
\begin{frame}{Why the representation theory is type A} 
\input{8diml_rep_slide1b.tex}
\end{frame}
%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%
\begin{frame}{Why the representation theory is type A} 
\input{8diml_rep_slide2.tex}
\end{frame}
%%%%%%%%%%%%%%%


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


\begin{itemize}	
	\item[[Da\brac] Z.\ Daugherty,   {\sl Degenerate two-boundary centralizer algebras} (to appear in Pac.\ J.\ Math., arXiv:1007.3950)
%	\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[[OR\brac] R.\ Orellana and A.\ Ram, \emph{
Affine braids, Markov traces and the category $\cO$}, 
{ Proceedings of the International Colloquium on Algebraic Groups and Homogeneous 
Spaces Mumbai 2004}, V.B.\ Mehta ed., Tata Institute of Fundamental Research, Narosa 
Publishing House, Amer.\ Math.\ Soc.\ (2007) 423--473.
\end{itemize}
In preparation: 
\begin{itemize} 
	\item[[Da2\brac] Z.\ Daugherty,   {\sl Centralizer properties of the graded Hecke algebra of type C}
%	\item [[DRV2\brac] Z.\ Daugherty, A.\ Ram, R.\ Virk, {\sl Affine and graded BMW algebras II} 
	\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}
