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

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

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

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



\def\brac{]}

\def\one{\mathbbm{1}}
\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}}



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      {\usebox\topalignbox}}}
\newcolumntype{M}{>{$\vcenter\bgroup\hbox\bgroup}c<{\egroup\egroup$}}



%%%%%%%%%%%%%%%%%
%\usepackage{etex} %fixes the fight that pictex has with every other drawing package
%\usepackage{pictexwd}
\usepackage{tikz}
%\input xy
%\xyoption{all}
%\xyoption{color}
\tikzstyle{V}=[draw, fill =black, circle, inner sep=0pt, minimum size=1.5pt]
\tikzstyle{bV}=[draw, fill =black, circle, inner sep=0pt, minimum size=4pt]
\tikzstyle{over}=[draw=white,double=black,line width=2pt, double distance=.75pt]
\tikzstyle{diagram}=[line width=.75pt, scale=\SCALE]

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

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


\newcounter{r}
\newcommand\Part[1]{
        \setcounter{r}{1}
	 \foreach \x in {#1}{
 	{\ifnum\value{r}=1
		\draw (0,\value{r}-1)--(\x,\value{r}-1); 
		\fi}
	\draw (0,\value{r}) to (\x,\value{r});
   	\foreach \y in {0, ..., \x} {\draw (\y,\value{r})--(\y,\value{r}-1);}
	\addtocounter{r}{1}
 }}


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

\def\UNIT{.25} %Sets the scale of your partitions 
%Whenever there's "yscale = - ...", that turns  the coordinates upside-down, to agree with how we count rows of partitions. 

%Self-contained tikz images for \Part and \fPart above. 
\newcommand{\PART}[1]{
\begin{tikzpicture}[xscale=\UNIT, yscale=-\UNIT] 
	\Part{#1}
\end{tikzpicture}
}
\newcommand{\fPART}[2]{
\begin{tikzpicture}[xscale=\UNIT, yscale=-\UNIT]
	\fPart{#1}{#2}
\end{tikzpicture}
}

 
 
 
%This is the lattice behind all the examples below. 
\def\Lattice{
%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (0) at (0,.4);
	\coordinate (11) at (0,1.2);
	\coordinate (21) at (-1,2);\coordinate (22) at (1,2);
	\foreach \x in {1, ..., 3}{\coordinate (3\x) at (-4+2*\x,3);}
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-6+2*\x,4.5);}
	\foreach \x in {1, ..., 7}{\coordinate (5\x) at (-8+2*\x,6);}
%edges in lattice:
	\draw (0)--(11)--(21) (11)--(22);
	\draw (21)--(31) (21)--(32) (22)--(32) (22)--(33);
	\draw (31)--(41) (31)--(42) (32)--(42)  (32)--(43) (32)--(44) (33)--(44)  (33)--(45);
	\draw (41)--(51) (41)--(52) (42)--(52)  (42)--(53)  (42)--(54)  (43)--(53)
		(43)--(55)  (44)--(54)  (44)--(55) (44)--(56) (45)--(56)  (45)--(57);
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (0) {$\emptyset$};
	\node at (11) {\PART{1}};
	\node at (22) {\PART{1,1}};
	\node at (21) {\PART{2}};
	\node at (33) {\PART{1,1,1}};
	\node at (32) {\PART{2,1}};
	\node at (31) {\PART{3}};
	\node at (45) {\PART{1,1,1,1}};
	\node at (44) {\PART{2,1,1}};
	\node at (43) {\PART{2,2}};
	\node at (42) {\PART{3,1}};
	\node at (41) {\PART{4}};
	\node at (57) {\PART{1,1,1,1,1}};
	\node at (56) {\PART{2,1,1,1}};
	\node at (55) {\PART{2,2,1}};
	\node at (54) {\PART{3,1,1}};
	\node at (53) {\PART{3,2}};
	\node at (52) {\PART{4,1}};
	\node at (51) {\PART{5}};
\end{scope}
\foreach \x in {-4,0,4} {\node at (\x , 6.75) {$\vdots$}; }
}


%
%\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{Representation theory and combinatorics of diagram algebras.}
\author[Zajj Daugherty]{Zajj Daugherty}
\date{May 15, 2016}




%%%%%DIAGRAM SLIDE%%%%%
\definecolor{TypeB}{rgb}{0, 0, 0.85}
\definecolor{TypeA}{rgb}{0.75, 0, .85}
\definecolor{TypeA2}{rgb}{0.85, 0, 0}
\definecolor{Univ}{rgb}{0, 0.65, 0.35}
\tikzstyle{overB}=[draw=white,double=TypeB,line width=1.5pt, double distance=.5pt]
\tikzstyle{overA}=[draw=white,double=TypeA,line width=1.5pt, double distance=.5pt]
\tikzstyle{overA2}=[draw=white,double=TypeA2,line width=1.5pt, double distance=.5pt]
\tikzstyle{overU}=[draw=white,double=Univ,line width=1.5pt, double distance=.5pt]
\def\OverB[#1,#2][#3,#4]{ %1,2=(x,y) position; 3,4=width,height
\draw[overB]   (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);
}
\def\OverA[#1,#2][#3,#4]{ %1,2=(x,y) position; 3,4=width,height
\draw[style=overA]   (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);
}
\def\OverAT[#1,#2][#3,#4]{ %1,2=(x,y) position; 3,4=width,height
\draw[overA2]   (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);
}
\def\OverU[#1,#2][#3,#4]{ %1,2=(x,y) position; 3,4=width,height
\draw[overU]   (#1,#2) .. controls ++(0,#4*.5-#2*.5) and ++(0,-#4*.5+#2*.5) .. (#3,#4);
}


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


\begin{frame}{Combinatorial representation theory}
\pause

\alert{Representation theory}: Given an algebra $A$\dots 
\begin{itemize}
\item What are the $A$-modules/representations?
\item What are the simple/indecomposable $A$-modules/reps? 
\item What is the action of the center of $A$?
\item What are their dimensions?
\item How can I combine modules to make new ones, and what are they in terms of the simple modules?
\end{itemize}\pause

In \alert{combinatorial} representation theory, we use combinatorial objects to index (construct a bijection to) modules and representations, and to encode information about them.

\end{frame}


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

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


%
%%%%%%%%%
%\begin{frame}{Diagram algebras and braids}
%The {\bf braid group}:
%
%$$\begin{tikzpicture}[line width=1pt]
%\node at (.25,0){};%PLACE MARKER
%\node at (1, 1.8) {\scriptsize \color{white} 1};%PLACE MARKER
%\node at (1, -.3) {\scriptsize \color{white} 1};%PLACE MARKER
%\node at (1, -2.4) {\scriptsize \color{white} 1};%PLACE MARKER
%\foreach \x in {1,...,5}{\node[bV] (t\x) at (\x,1.5){};\node[bV] (b\x) at (\x,0){};}
%%%%%%%%%%%%%%
%	\draw[over] (b2)--(t1); 
%	\draw[over] (b4)--(t2); 
%	\draw[over](b1)--(t3); 
%	\draw[over] (b5)--(t4); 
%	\draw[over] (b3)--(t5); 
%%%%%%%%%%%%%%
%\pgftransformyshift{-2.05cm}
%%%%%%%%%%%%%%
%\foreach \x in {1,...,5}{\node[bV] (bt\x) at (\x,1.5){};\node[bV] (bb\x) at (\x,0){};}
%	\draw[over] (bb2)--(bt1); 
%	\draw[over] (bb1)--(bt2); 
%	\draw[over] (bb4)--(bt3); 
%	\draw[over] (bb5)--(bt4); 
%	\draw[over] (bb3)--(bt5); 
%%%%%%%%%%%%%%
%	\foreach \x in {1,...,5}{
%	\draw[dotted, alert] (b\x)--(bt\x);
%	}
%%%%%%%%%%%%%%
%\pgftransformyshift{1cm}
%\pgftransformxshift{6cm}
%%%%%%%%%%%%%%
%\uncover<2->{
%	\node at (0,.75) {$=$};
%	\draw (2,1.5).. controls (4,1) .. (5,0);
%	\Under[1,1.5][1.75,.75];
%	\Under[1.25,.75][2,0];
%	\Under[4,1.5][4.25,.75];
%	\Over[3,1.5][1.25,.75];
%	\Over[5,1.5][4,.75];
%	\Over[1.75,.75][1,0];
%	\Over[4,.75][4,0];
%	\Over[4.25,.75][3,0];
%\foreach \x in {1,...,5}{\node[bV] (rt\x) at (\x,1.5){};\node[bV] (rb\x) at (\x,0){};}
%}
%\end{tikzpicture}$$
%(with multiplication given by concatenation)
%\end{frame}
%
%%%%%%%%%
%\begin{frame}{Diagram algebras and braids}
%The {\bf affine (one-pole) braid group}:
%
%$$\begin{tikzpicture}[line width=1pt]
%\node at (1, 1.8) {\scriptsize \color{white} 1};
%\node at (1, -.3) {\scriptsize \color{white} 1};
%\node at (1, -2.4) {\scriptsize \color{white} 1};
%%%%%%%%%%%%%%
%\draw[dotted, alert] (.35,-.1)--(.35,-.4) (.5,-.1)--(.5,-.4);
%\PoleTop[\TOP]
%\foreach \x in {1,...,5}{\node[bV] (t\x) at (\x,1.5){};\node[bV] (b\x) at (\x,0){};}
%	\draw[over] (b2)--(t1); 
%	\draw[over] (b4)--(t2); 
%	\draw[over] (b5)--(t4); 
%	\draw[over] (b3)--(t5); 
%\PoleBottom[0]
%	\Over[.25,.5][1,0] \node[bV] at (b1){};
%	\Pole[0][\TOP]
%	\Over[3,1.5][.25,.5] \node[bV] at (t3){};
%%%%%%%%%%%%%%
%\pgftransformyshift{-2.05cm}
%%%%%%%%%%%%%%
%\foreach \x in {1,...,5}{\node[bV] (bt\x) at (\x,1.5){};\node[bV] (bb\x) at (\x,0){};}
%	\PoleTop[\TOP]
%	\PoleBottom[0]
%	\Pole[0][\TOP]
%	\draw[over] (bb2)--(bt1); 
%	\draw[over] (bb1)--(bt2); 
%	\draw[over] (bb4)--(bt3); 
%	\draw[over] (bb5)--(bt4); 
%	\draw[over] (bb3)--(bt5); 
%%%%%%%%%%%%%%
%	\foreach \x in {1,...,5}{
%	\draw[dotted, alert] (b\x)--(bt\x);
%	}
%%%%%%%%%%%%%%
%\pgftransformyshift{1cm}
%\pgftransformxshift{6cm}
%%%%%%%%%%%%%%
%\node at (-.25,.75){$=$};
%\PoleTop[\TOP]
%\PoleBottom[0]
%	\draw (2,1.5).. controls (4,1) .. (5,0);
%	\Under[1,1.5][1.75,.75];
%	\Under[4,1.5][4.25,.75];
%	\Under[.25,.75][2,0];
%	\Pole[0][\TOP]
%	\Over[3,1.5][.25,.75];
%	\Over[5,1.5][4,.75];
%	\Over[1.75,.75][1,0];
%	\Over[4,.75][4,0];
%	\Over[4.25,.75][3,0];
%	\foreach \x in {1,...,5}{
%		\node[bV] (rt\x) at (\x,1.5){};\node[bV] (rb\x) at (\x,0){};
%	 }
%\end{tikzpicture}$$
%(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}$\begin{tikzpicture}[line width=1pt]
%\PoleTop[\TOP]
%\PoleBottom[0]
%	\draw (2,1.5).. controls (4,1) .. (5,0);
%	\Under[1,1.5][1.75,.75];
%	\Under[.25,.75][2,0];
%	\Pole[0][.5*\TOP]
%	\Under[4,1.5][4.25,.75];
%	\Pole[.5*\TOP][\TOP]
%	\Over[3,1.5][.25,.75];
%	\Over[5,1.5][4,.75];
%	\Over[1.75,.75][1,0];
%	\Over[4,.75][4,0];
%	\Over[4.25,.75][3,0];
%	\foreach \x in {1,...,5}{
%		 \filldraw [black] (\x,1.5) circle (2pt);
%		 \filldraw [black] (\x,0) circle (2pt);
%	 }
%\end{tikzpicture}$\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}
%



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

\end{enumerate}
\end{frame}

\begin{frame}{{Motivating example: Schur-Weyl Duality}}
{} {Schur (1901):} {\small $S_k$ and $\GL_n$ have commuting actions on $\left( \CC^n\right)^{\otimes k}$.}\\
 Even better,
$$\!\!\!\underbrace{\End_{\GL_n}\left(\left( \CC^n\right)^{\otimes k} \right)}_{\text{\scriptsize (all linear maps that} \atop \text{\scriptsize commute with $\GL_n$)}} 
	= \underbrace{\pi(\CC S_k)\phantom{\Big|}\!\!}_{\text{\scriptsize (img of $S_k$} \atop\text{\scriptsize action)}}
	 \quad \text{ and } \quad \End_{S_k}\left( \left( \CC^n\right)^{\otimes k}\right) 
	 = \underbrace{\rho(\CC \GL_n)\phantom{\Big|}\!\!}_{\text{\scriptsize (img of $\GL_n$} \atop\text{\scriptsize action)}}
\!\!. $$%
\uncover<2->{\begin{block}{\bf Why this is exciting:}
		The double-centralizer relationship produces
			$$\left( \CC^n\right)^{\otimes k}  \cong \bigoplus_{\tiny \lambda \vdash k }  G^\lambda \otimes S^\lambda
			\quad \text{ as a $\GL_n$-$S_k$ bimodule,}$$
where 		\begin{tabular}{rcl}
			$G^\lambda$ &are distinct irreducible& $\GL_n$-modules\\
			$S^\lambda$ &are distinct irreducible& $S_k$-modules\\
			\end{tabular}
	\end{block}}
	
	\uncover<3->{
	For example, 
	\centerline{$\CC^n \otimes \CC^n \otimes \CC^n  
	= \Big(G^{\PART{3}} \otimes S^{\PART{3}}\Big) 
	\oplus  \Big(G^{\PART{2,1}} \otimes S^{\PART{2,1}}\Big) 
	\oplus  \Big(G^{\PART{1,1,1}} \otimes S^{\PART{1,1,1}}\Big)$}}
	
\end{frame}





\begin{frame}{Representation theory of $V^{\otimes k}$}

\centerline{$V = \CC = L(\PART{1})  \uncover<2->{, \qquad L(\PART{1}) }\uncover<3->{\otimes L(\PART{1}) }\uncover<4->{\otimes L(\PART{1}) }\uncover<5->{\otimes L(\PART{1}) }\uncover<6->{\otimes L(\PART{1})  \cdots}$}



\centerline{\uncover<2->{\begin{tikzpicture}[xscale=3*\UNIT, yscale=-4.5*\UNIT]
\begin{scope}[black]%If you want to gray out the rest of the lattice
	%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (0) at (0,.4);
	\coordinate (11) at (0,1.2);
	\coordinate (21) at (-1,2);\coordinate (22) at (1,2);
	\foreach \x in {1, ..., 3}{\coordinate (3\x) at (-4+2*\x,3);}
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-6+2*\x,4.5);}
	\foreach \x in {1, ..., 7}{\coordinate (5\x) at (-8+2*\x,6);}
%edges in lattice:
	\draw (0)--(11);
	\uncover<3->{\draw (11)--(21) (11)--(22);}
	\uncover<4->{\draw (21)--(31) (21)--(32) (22)--(32) (22)--(33);}
	\uncover<5->{\draw (31)--(41) (31)--(42) (32)--(42)  (32)--(43) (32)--(44) (33)--(44)  (33)--(45);}
	\uncover<6->{\draw (41)--(51) (41)--(52) (42)--(52)  (42)--(53)  (42)--(54)  (43)--(53)
		(43)--(55)  (44)--(54)  (44)--(55) (44)--(56) (45)--(56)  (45)--(57);}
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (0) {$\emptyset$};
	\node at (11) {\PART{1}};
	\uncover<3->{
		\node at (22) {\PART{1,1}};
		\node at (21) {\PART{2}};}
	\uncover<4->{
		\node at (33) {\PART{1,1,1}};
		\node at (32) {\PART{2,1}};
		\node at (31) {\PART{3}};}
	\uncover<5->{
		\node at (45) {\PART{1,1,1,1}};
		\node at (44) {\PART{2,1,1}};
		\node at (43) {\PART{2,2}};
		\node at (42) {\PART{3,1}};
		\node at (41) {\PART{4}};}
	\uncover<6->{
		\node at (57) {\PART{1,1,1,1,1}};
		\node at (56) {\PART{2,1,1,1}};
		\node at (55) {\PART{2,2,1}};
		\node at (54) {\PART{3,1,1}};
		\node at (53) {\PART{3,2}};
		\node at (52) {\PART{4,1}};
		\node at (51) {\PART{5}};}
\end{scope}
	\uncover<6->{\foreach \x in {-4,0,4} {\node at (\x , 6.75) {$\vdots$}; }}
\end{scope} 
\begin{scope}[every node/.style={fill=white}]
\uncover<8->{	\node (a) at (0) {\color{blue!50!red!80!black}$\emptyset$};
	\node (b) at (11) {\fPART{1}{blue!50!red!40}};
	\node (c) at (21) {\fPART{2}{blue!50!red!40}};
	\node (c') at (22) {\fPART{1,1}{blue!50!red!40}};
	\node (d') at (32) {\fPART{2,1}{blue!50!red!40}};
	\node (d) at (31) {\fPART{3}{blue!50!red!40}};
	\node (e) at (42) {\fPART{3,1}{blue!50!red!40}};
	\node (e') at (43) {\fPART{2,2}{blue!50!red!40}};}
\uncover<7->{	\node (f) at (53) {\fPART{3,2}{blue!40}};}
\end{scope}
\uncover<8->{\begin{scope}[blue!50!red!80!black, thick, ->]
\draw (a)--(b); \draw (b)--(c);\draw (b)--(c'); \draw (c)--(d);  \draw (c')--(d'); \draw (c)--(d'); \draw (d)--(e); \draw (d')--(e); \draw (d')--(e'); \draw (e)--(f);\draw (e')--(f);
\end{scope}}
\end{tikzpicture}}}




 \end{frame} 





\begin{frame}{More centralizer algebras}

\only<1,7,8>{\hbox{
\hspace{-.1in}
\begin{minipage}[t]{2in}
Brauer (1937) \\
\small Orthogonal and symplectic groups \only<1->{(and Lie algebras)} acting on $(\CC^n)^{\otimes k}$ diagonally
centralize\\ the \textbf{Brauer algebra}:\\
\begin{tikzpicture}[line width=1pt, scale=.75]
	\foreach\x in {1,...,5}{
		\node[bV] (t\x) at (\x,1){};
		\node[bV] (b\x) at (\x,0){};}
	\draw [bend left] (t2) to (t1) (b2) to (b3);
	\draw (b1)--(t3) (b5)--(t4) (b4)--(t5);
	\foreach\x/\y in {1/a,2/b,3/c,4/d,5/e}{\node[below] at (b\x) {\small\alert{$v_{\y}$}};}
	\node[above] at (t1) {\small\alert{$v_i$}};
	\node[above] at (t2) {\small\alert{$v_i$}};
	\node[above] at (t3) {\small\alert{$v_a$}};
	\node[above] at (t4) {\small\alert{$v_d$}};
	\node[above] at (t5) {\small\alert{$v_d$}};
	\foreach\x in {1,...,4}{
		\node[above] at (\x+.5,1) {\small\alert{$\otimes$}};
		\node[below] at (\x+.5,0) {\small\alert{$\otimes$}};}
	\node[left] at (1,1.4) {\small\alert{$\displaystyle \delta_{b,c} \sum_{i=1}^n$}};
	\draw[|->,alert] (5.5,0)--(5.5,1);
	\foreach\x in {1,...,5}{\node[bV] at (t\x){};\node[bV] at (b\x){};}
\end{tikzpicture}

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

\centerline{with 
$\begin{matrix}\begin{tikzpicture}[line width=1pt, scale=.75]
	\draw (0,0) ellipse (15pt and 7pt);
\end{tikzpicture}\end{matrix}
	= 2$}
\end{minipage}}

~\\~\\

\uncover<8>{\textbf{Either way:}\\
Diagrams encoding maps $V^{\otimes k} \to V^{\otimes k}$ that commute with the action of some classical algebra. }}%
\only<2,3,4,5,6>{{Rep theory of $V^{\otimes k}$, orthogonal and symplectic:}

\centerline{$V = \CC = L(\PART{1})  \uncover<3->{, \qquad L(\PART{1}) }\uncover<4->{\otimes L(\PART{1}) }\uncover<5->{\otimes L(\PART{1}) }\uncover<6->{\otimes L(\PART{1})  \cdots}$}



\centerline{\uncover<3->{\begin{tikzpicture}[xscale=3*\UNIT, yscale=-4.5*\UNIT]
\begin{scope}[black]%If you want to gray out the rest of the lattice
	%Sets the coordinates for the partitions of 0 to 5:
	\coordinate (0) at (0,.4);
	\coordinate (11) at (0,1.2);
	\coordinate (2b1) at (-2,2);\coordinate (21) at (0,2);\coordinate (22) at (2,2);
	\foreach \x in {1, ..., 3}{\coordinate (3\x) at (-3+2*\x,3);}\coordinate (3b1) at (-3,3);
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-4+2*\x,5);}\foreach \x in {1,2}{\coordinate (4b\x) at (-8+2*\x,5);}
%edges in lattice:
	\draw (0)--(11);
	\uncover<4->{\draw (11)--(21) (11)--(22) (11)--(2b1);}
	\uncover<5->{\draw (21)--(31) (21)--(32) (22)--(32) (22)--(33) (21)--(3b1) (22)--(3b1) (2b1)--(3b1);}
	\uncover<6->{\draw (31)--(41) (31)--(42) (32)--(42)  (32)--(43) (32)--(44) (33)--(44)  (33)--(45)
				(3b1) to (4b1) (3b1) to (4b2) (31) to (4b1) (32) to (4b1) (32) to (4b2) (33) to (4b2);}
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	\node at (0) {$\emptyset$};
	\node at (11) {\PART{1}};
	\uncover<4->{
		\node at (22) {\PART{1,1}};
		\node at (21) {\PART{2}};
		\node at (2b1) {$\emptyset$};}
	\uncover<5->{
		\node at (3b1) {\PART{1}};
		\node at (33) {\PART{1,1,1}};
		\node at (32) {\PART{2,1}};
		\node at (31) {\PART{3}};}
	\uncover<6->{
		\node at (45) {\PART{1,1,1,1}};
		\node at (44) {\PART{2,1,1}};
		\node at (43) {\PART{2,2}};
		\node at (42) {\PART{3,1}};
		\node at (41) {\PART{4}};
		\node at (4b2) {\PART{1,1}};
		\node at (4b1) {\PART{2}};}
\end{scope}
	\uncover<6->{\foreach \x in {-4,0,4} {\node at (\x , 5.75) {$\vdots$}; }}
\end{scope} 
\end{tikzpicture}}}}




 \end{frame} 



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

$\cU\otimes \cU$ has an invertible element $\cR=\sum_{\cR} R_1\otimes R_2$ that yields a map\\
\centerline{\begin{tikzpicture}[scale=.8, line width=1pt ]
	\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  action on $V\otimes W$\\
for any $\cU$-module $V$. \pause

~\\

The \only<4>{\alert{one-pole/affine} }\only<5>{\alert{two-pole} }braid group shares
 a commuting action\\ with $\cU$ on $\only<4->{\alert{M \otimes}}V^{\otimes k}\only<5->{\alert{\otimes N}}$:
$$\begin{matrix}\begin{tikzpicture}[line width=1pt]
	\uncover<4>{
		\PoleTop[\TOP]
		\PoleBottom[0]}
	\uncover<5>{
		\SymPolesTop[\TOP][5]
		\SymPolesBottom[0][5]}
	\draw (2,1.5).. controls (4,1) .. (5,0);
	\Under[1,1.5][1.75,.75];
	\uncover<3-4>{\Under[4,1.5][4.5,.75];}
	\uncover<5>{\Under[4,1.5][5.75,.75];}
	\uncover<3>{
		\Under[1.25,.75][2,0];
		\Over[3,1.5][1.25,.75];}
	\uncover<4->{
		\Under[.25,.75][2,0];
		\Pole[0][\TOP]
		\Over[3,1.5][.25,.75];}
	\uncover<5>{\RRPole[0,\TOP][5]}
	\Over[5,1.5][4,.75];
	\Over[1.75,.75][1,0];
	\Over[4,.75][4,0];
	\uncover<3-4>{\Over[4.5,.75][3,0];}
	\uncover<5>{\Over[5.75,.75][3,0];}
\foreach \x in {1,...,5}{
	\node[bV, label=above:{\alert{$V$}}] (t\x) at (\x,1.5){};
	\node[bV, label=below:{\alert{$V$}}] (b\x) at (\x,0){};
	\ifnum \x<5 
		\node at (\x+.5,1.821) {\alert{$\otimes$}};
		\node at (\x+.5,-.357) {\alert{$\otimes$}}; \fi
	}
\uncover<4->{
	\node at (.55,1.823) {\alert{$M \otimes$}};
	\node at (.55,-.355) {\alert{$M \otimes$}};}
\uncover<5>{
	\node at (5.45,1.823) {\alert{$\otimes N$}};
	\node at (5.45,-.355) {\alert{$\otimes N$}};}
\end{tikzpicture}\end{matrix}
\qquad \quad
\uncover<4->{\begin{matrix}\text{\scriptsize Around the pole:}\\
\begin{matrix}	\begin{tikzpicture}[line width=1pt]
	\Under[.8,.9][.2,.5]
	\Pole[0.2][0.8]
	\PoleTop[0.8]
	\PoleBottom[0.2]
	\Over[.8,.1][.2,.5]
	\node[V] at (.8,.9){}; \node[V] at (.8,.1){}; 
	\node[above] at (.6,.9) {\tiny$M \!\!\otimes\! V$}; 
	\node[below] at (.6,.1) {\tiny$M \!\!\otimes\! V$}; 
	\end{tikzpicture}
\end{matrix}
=  \check R_{MV} \check R_{VM} \end{matrix}}
$$

\end{frame}


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

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

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

\begin{frame}%TABLE OF DIAGRAM ALGEBRAS BMW
\hbox{\hspace{-.4in}\includegraphics{diagrams-slide-BMW}}
\end{frame}
%
%
%\begin{frame}%TABLE OF DIAGRAM ALGEBRAS Hecke
%\hbox{\hspace{-.4in}\includegraphics[width=5in]{diagrams-slide-Hecke}}
%\end{frame}


\begin{frame}
\hbox{\hspace{-.4in}
 \begin{tabular}{MM}
    \includegraphics{diagrams-slide-BMWSide}&
\begin{minipage}{2.75in}\small\pause

{\bf Nazarov (95):} \alert{degenerate affine BMW algebras}

\centerline{	\begin{tikzpicture}[scale=.75]\node at (1,1.2) {};
		\foreach \x in {1, ..., 5}{
		\node[V] (b\x) at (\x*.75+.25, 0){};	\node[V] (t\x) at (\x*.75+.25, 1) {};}
	\draw (b2) to [bend left=45] (b4) (t4)--(b5);
	\draw (t1)--(b3)  (t2)--(b1) (t3) to [bend right=45] (t5);
	\node[draw, fill =alert, circle, inner sep=0pt, minimum size=3pt] at (1.3, .4){};
	\draw (6,.8)--(6,.2) to [bend right=75] (6.75,.2)--(6.75,.8) to [bend right=75] (6,.8);
	\foreach \y in {.25,.5,.75}{
	\node[draw, fill =alert, circle, inner sep=0pt, minimum size=3pt] at (6, \y){};}
	\draw[|-|] (5.8, .1) to node[midway,left]{\tiny $\ell$} (5.8,.9);
	\node[right] at (6.75,.5) {\footnotesize $=z_\ell \in \CC$};
	\end{tikzpicture}}

act on $M\otimes V^{\otimes k}$, commuting with the action of the Lie algebras of types B, C, D. 

\medskip\pause

{\bf H\"aring-Oldenburg (98) and Orellana-Ram (04):} \alert{affine BMW algebras} act on $M\otimes V^{\otimes k}$, commuting with the action of the quantum groups of types B, C, D.

\medskip\pause

\alert{\bf Centralizer perspective:} (D.-Ram-Virk) Use centralizer relationships to study these the affine and degenerate affine algebras simultaneously (representation theory of the quantum groups and the Lie algebras are basically the same). \\
Some results:\\
\alert{(a)} The center of each algebra.\\
\alert{(b)}  Difficult ``admissibility conditions'' handled.\\
\alert{(c)}  Powerful ``intertwiner'' operators.


\end{minipage}\end{tabular}}

\end{frame}


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


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


\begin{frame}\small

\centerline{\includegraphics{diagrams-slide-FR}}
\centerline{\includegraphics{diagrams-slide-LR}}\pause


\alert{\bf Two boundary algebras:}\\
{\bf Mitra, Nienhuis, De Gier, Batchelor (2004)}: Studying the six-vertex model with additional integrable boundary terms, introduced the \alert{two-boundary Temperley-Lieb algebra} $TL_k$:

\centerline{\begin{tikzpicture}[scale=.75]
	\foreach \x in {1, ..., 5}{
		\node[draw, fill =black, circle, inner sep=0pt, minimum size=3pt] (b\x) at (\x*.75+.25, 0){};	
		\node[draw, fill =black, circle, inner sep=0pt, minimum size=3pt] (t\x) at (\x*.75+.25, 2) {};}
	\draw[line width=2.5pt] (.5,-.1)--(.5,2.1);
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (L1) at (.5,.4) {}; 
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (L2) at (.5,.8) {}; 
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (L3) at (.5,1.2) {}; 
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (L4) at (.5,1.6) {}; 
	\draw[line width=2.5pt] (4.5,-.1)--(4.5,2.1);
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (R1) at (4.5,.4) {}; 
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (R2) at (4.5,.8) {};
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (R3) at (4.5,1.2) {}; 
	\node[draw, fill =white, circle, inner sep=0pt, minimum size=3.5pt] (R4) at (4.5,1.6) {};
	\draw[line width=1pt] (t1) to [bend right] (t2) (t3) to [bend left] (L4)  
		(L1) to [bend left] (b1) (t4) to [bend right] (R3) 
		(t5) to [bend right] (R4) 
		(b2) to [bend left] (b5) (b3)  to [bend left]  (b4)
		(L3) to (R2) (L2) to (R1);
	\draw[|-|] (.75,-.3) to node[midway,below]{\tiny $k$ dots} (4.25,-.3);
	\draw[|-|] (0.2,.2) to node[midway, sloped, above]{\tiny even $\#$ dots} (0.2,1.8);
	\node at (2.5,2.3) {\tiny non-crossing diagrams};
	\end{tikzpicture}}\pause

{\bf De Gier, Nichols (2008)}: Explored representation theory of $TL_k$ using diagrams and established a connection to the affine Hecke algebras of type A and C.   

%\pause

%{\bf D. (2010)}: The centralizer of $\fgl_n$ acting on tensor space $M \otimes V^{\otimes k} \otimes N$ displays type C combinatorics for good choices of $M$, $N$, and $V$. 
 


\end{frame}


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


\begin{frame}{Affine type C Hecke algebra and two-boundary braids}
\def\SCALE{1.2}\small
\centerline{$\begin{tikzpicture}[diagram]
\node at (7,.3) {\tiny {$k$}};
	\node at (0,.3) {\tiny \alert<5>{$0$}};
	\node at (1,.3) {\tiny \alert<5>{$1$}};
	\node at (2,.3) {\tiny \alert<4>{$2$}};
	\node at (3,.3) {\tiny \alert<4>{$3$}};
	\node at (4,.3) {\tiny {$4$}};
\node at (5,.3) {\tiny {$k\!-\!2$}};
\node at (6,.3) {\tiny {$k\!-\!1$}};
\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<5>{\draw[thick,structure, double, double distance=2pt] (0,0)--(1,0);
	\filldraw[white] (0,0) circle (4pt);
	\filldraw[white] (1,0) circle (4pt);
	\draw[thick,structure] (0,0) circle (4pt);
	\draw[thick,structure] (1,0) circle (4pt);
}
\only<1-4>{\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<4>{
	\draw[thick, structure] (2,0)--(3,0);
	\filldraw[white] (2,0) circle (4pt);
	\filldraw[white] (3,0) circle (4pt);
	\draw[thick,structure](3,0) circle (4pt);
	\draw[thick,structure] (2,0) circle (4pt);
}
\end{tikzpicture}$}

\def\SCALE{.75}
Fix  constants $t_0, t_{k},$ and $t=t_1=\cdots=t_{k-1}$. The \alert{affine Hecke algebra of type C},  $\cH_k$, is generated by $T_0, T_1, \dots, T_{k}$ with relations 
\only<1>{$$\underbrace{T_i T_j \dots }_{m_{i,j} \text{ factors}} =  \underbrace{T_j T_i \dots }_{m_{i,j} \text{ factors}}
\qquad  \text{ where }
 \qquad  m_{i,j} =  \quad \begin{matrix}
\text{\small$2$ \quad  if } \quad 
	 	\begin{minipage}[t]{1cm}
		\begin{tikzpicture}[diagram]
			\filldraw[white] (1,0) circle (4pt);
			\draw[thick] (1,0) circle (4pt);
			\node at (1,.4) {\tiny {$i$}};
			\filldraw[white] (2,0) circle (4pt);
			\draw[thick] (2,0) circle (4pt);
			\node at (2,.4) {\tiny {$j$}};
		\end{tikzpicture}\end{minipage}\\ 
\text{\small$3$ \quad  if } \quad 
	 	\begin{minipage}[t]{1cm}
		\begin{tikzpicture}[diagram]
			\draw[thick] (1,0)--(2,0);
			\filldraw[white] (1,0) circle (4pt);
			\draw[thick] (1,0) circle (4pt);
			\node at (1,.4) {\tiny {$i$}};
			\filldraw[white] (2,0) circle (4pt);
			\draw[thick] (2,0) circle (4pt);
			\node at (2,.4) {\tiny {$j$}};
		\end{tikzpicture}\end{minipage}\\
\text{\small$4$ \quad  if } \quad 
	 	\begin{minipage}[t]{1cm}
		\begin{tikzpicture}[diagram]
		\draw[thick, double, double distance=2pt] (1,0)--(2,0);
			\filldraw[white] (1,0) circle (4pt);
			\draw[thick] (1,0) circle (4pt);
			\node at (1,.4) {\tiny {$i$}};
			\filldraw[white] (2,0) circle (4pt);
			\draw[thick] (2,0) circle (4pt);
			\node at (2,.4) {\tiny {$j$}};
		\end{tikzpicture}\end{minipage}\end{matrix}
$$%
and $T_i^2 = (t_i^{1/2} - t_i^{-1/2}) T_i + 1.$}
\only<2->{$$\underbrace{T_i T_j \dots }_{m_{i,j} \text{ factors}} =  \underbrace{T_j T_i \dots }_{m_{i,j} \text{ factors}}
\qquad  \text{ and }
 \qquad  T_i^2 = (t_i^{1/2} - t_i^{-1/2}) T_i + 1.
$$}

\pause\pause

\def\SCALE{.6}
The \alert{two-boundary (two-pole) braid group} $B_k$ is generated by 
\centerline{\def\TOP{1}\def\K{1}
\begin{tikzpicture}[diagram]
\node at (0,.5*\TOP) {$T_k=$};
%%%%%%%%%%%%%%
\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]
\RRPole[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]
	\Pole[0][1]
	\Over[1,0][.2,.5]
	\Nodes[0][\K]
	\Nodes[\TOP][\K]
%%%%%%%%%%%%%
\pgftransformxshift{4 cm}
%%%%%%%%%%%%%
\node[left] at (1,.5*\TOP) {and \quad $T_i=$};
%%%%%%%%%%%%%%
%\pgftransformxshift{1.2cm}
%%%%%%%%%%%%%%
	\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]
%%%%%%%%%%%%%
\pgftransformxshift{2 cm}
\pgftransformxshift{.25 cm}
%%%%%%%%%%%%%
\node[right]  at (0, .5*\TOP)  {\qquad for $1\leq i \leq k-1$.};
\end{tikzpicture}
}


\uncover<4->{\def\SCALE{.6}
\alert{Relations:}\\
\centerline{$
{\def\TOP{2.5}\begin{tikzpicture}[diagram]
	\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}}$}
}%
\uncover<5->{
\centerline{${\def\TOP{2.5}\begin{tikzpicture}[diagram]
\node[left] at (-.5,.5*\TOP) {$T_1 T_0 T_1 T_0~=$};
\PoleTop[\TOP]
	\PoleBottom[0]
	\Under[.2,.5][2,1.5]
	\Under[0,1.5][2,2.5]
	\Pole[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]
	\Pole[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) {$= ~ T_0 T_1 T_0 T_1$};
\end{tikzpicture}}$}}
\end{frame}



\begin{frame}{Affine type C Hecke algebra and two-boundary braids}
Punchline:
\begin{itemize}
\item For any for any complex reductive Lie algebras $\fg$, the quantum group $\cU_q \fg$ and the two-boundary braid group $B_k$  have commuting actions on  $M \otimes (V)^{\otimes k} \otimes N$.
\item When $\fg = \fgl_n$, for good choices of  $M$, $N$, and $V$, the action of the  two-boundary braid group factors to an action of 
the affine Hecke algebra of type $C$.
\end{itemize}

\pause

Some consequences:
\begin{enumerate}[(a)]
\item A combinatorial classification and construction of irreducible representations of $H_k$ (type C with distinct parameters).
\item A diagrammatic intuition for $H_k$.
\item A classification of the representations of $TL_k$ via the action of its center. 
\end{enumerate}


\end{frame}


%\begin{frame}\large
%\centerline{Thanks! }
%
%\bigskip
%
%\small
%
%\alert{[Da10\brac} \emph{Degenerate two-boundary centralizer algebras}, Pacific J. Math., 258-1 (2012) 91--142.
%
%
%\medskip
%
%\alert{[DRV14\brac} \emph{Affine and degenerate affine BMW algebras: the center}, with Arun Ram and Rahbar Virk, to appear in to appear in Osaka J. Math., 51-1 (2014).
%
%\medskip
%
%\alert{[DRV13\brac} \emph{Affine and degenerate affine BMW algebras: actions on tensor space}, with Arun Ram and Rahbar Virk, Selecta Math., 19-2 (2013) 611--653.
%
%
%\medskip
%
%\alert{[DR\brac} \emph{Two boundary Hecke Algebras and the combinatorics of type $(C_n^{\vee}, C)$ Hecke algebras}, with Arun Ram (in progress).
%
%\end{frame}

\begin{frame}%TABLE OF DIAGRAM ALGEBRAS C
\hbox{\hspace{-.4in}\includegraphics[width=5in]{diagrams-slideC}}
\end{frame}

\end{document}