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%%% Characters %%%------------------------------------------------------------
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\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}}

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\def\dim{\mathrm{dim}} 
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\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}} 
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\def\vep{\varepsilon}



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}
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\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][#3]{%1=pole locations, 2=top, 3=k 
	\foreach\x in {#1}{
		\draw (\x+.1,#2) -- (\x+.1,#2+.15) (\x-.1,#2) -- (\x-.1,#2+.15) ;
		\draw (\x+.1,#2+.15) arc (0:360:1mm and .5mm);}
	%Nodes 
	\foreach \x in {1,...,#3} {\draw (\x,#2)  to (\x,#2+.05) node[V]{};}
	}
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	\foreach\x in {#1}{
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	}
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	\Tops[#1][#3][#4]
	\Bottoms[#1][#2][#4]
	}
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	\node at (#3,#1-.4) {#4};		}
	
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%Self-contained tikz images for \Part above. 
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\def\RBIG{\color{white}\right|\color{black}}
\def\XNG{\color{white}$\bullet$}


\title{Representation theory and combinatorics \\
of braid algebras and their quotients.}
\author[Zajj Daugherty]{Zajj Daugherty}
\date{January 14, 2020}


\newcommand{\GREEN}[1]{{\color{Univ}#1}}

%%%%%DIAGRAM SLIDE%%%%%
\definecolor{TypeB}{rgb}{0, .4, 1}
\definecolor{TypeA}{rgb}{0.6, 0, 1}
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\tikzstyle{overU}=[draw=white,double=Univ,line width=1.5pt, double distance=.5pt]
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}
\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}{Schur-Weyl Duality}
The {\bf symmetric group} $S_k$ (permutations) as diagrams:

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




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

\end{enumerate}
\end{frame}

\begin{frame}{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}{Powerful consequence: a duality between representations}
		The double-centralizer relationship produces
			$$\left( \CC^n\right)^{\otimes k}  \cong \bigoplus_{\tiny \lambda \vdash k }  G^\lambda \otimes S^\lambda
			\quad \text{ as a $\GL_n$-$S_k$ bimodule,}$$
where 		\begin{tabular}{rcl}
			$G^\lambda$ &are distinct irreducible& $\GL_n$-modules\\
			$S^\lambda$ &are distinct irreducible& $S_k$-modules\\
			\end{tabular}
	\end{block}}
	
	\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}{More centralizer algebras}

\hbox{
\hspace{-.1in}
\begin{minipage}[t]{2in}
Brauer (1937) \\
\small Orthogonal and symplectic groups \only<3->{(and Lie algebras)} acting on $(\CC^n)^{\otimes k}$ diagonally\only<1-2>{\\}
centralize\only<3->{\\} the \textbf{Brauer algebra}:\\\only<1-2>{~\\}
\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[|->,themecolor] (5.5,0)--(5.5,1);
	\foreach\x in {1,...,5}{\node[bV] at (t\x){};\node[bV] at (b\x){};}
\end{tikzpicture}

\centerline{with 
$\begin{matrix}\begin{tikzpicture}[line width=1pt, scale=.75]
	\draw (0,0) ellipse (15pt and 7pt);
\end{tikzpicture}\end{matrix}
	= n$}
\end{minipage}\qquad
\uncover<2->{\begin{minipage}[t]{2.1in}
Temperley-Lieb (1971) \\
\small $\GL_2$ and $\SL_2$ \only<3->{(and $\fgl_2$ and $\fsl_2$)} acting on $(\CC^2)^{\otimes k}$\only<1-2>{\\}
diagonally centralize\only<3->{\\}
 the\only<1-2>{\\} \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[|->,themecolor] (5.5,0)--(5.5,1);
	\foreach\x in {1,...,5}{\node[bV] at (t\x){};\node[bV] at (b\x){};}
\end{tikzpicture}

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

~\\~\\

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

\end{frame}


\begin{frame}{Quantum groups and braids}
\small
Let $\fg$ be a Lie algebra, and fix $q \in \CC$.\\
 One deformation of $\fg$ is the Drinfel'd-Jimbo \alert{quantum group} $\cU=\cU_q\fg$. \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,  ]
	\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 $V^{\otimes k}$\\
for any $\cU$-modules $V, W$. \pause

~\\

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

\end{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}
\centerline{\hbox{\setlength{\tabcolsep}{3pt} 
\def\arraystretch{.7}\begin{tabular}{M@{\hspace{-.1in}}rMMMM@{\hspace{-.1in}}M}
\multirow{3}{*}{
      \begin{tikzpicture}
		\draw[white](0,0)  to 
			node[midway,black, TypeB] {\footnotesize \begin{tabular}{c}Orthogonal\\and\\symplectic\\
	\scriptsize(types B, C, D)\end{tabular}}  (0,3.75);
	\end{tikzpicture}
	}
 &&\footnotesize$V^{\otimes k}$ & \footnotesize$M \otimes V^{\otimes k}$ & \footnotesize$M \otimes V^{\otimes k} \otimes N$\\
%%%%%%%%%%%%%%%%%%%
&\footnotesize Qu.\ grps:& \tikz{
%	\node at (0,0) {};
	\node[TypeB, draw,  fill=white, line width=1.5pt,  align=center]  at (0,0) {
	\footnotesize BMW algebra\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
%	\node at (2,1.5){};
	\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x, 0){};	\node[V,TypeB] (t\x) at (\x, 1.25) {};}
	\draw (t1)--(b3)  (t3) to [bend right] (t5);
	\draw[overB]
			(t2)--(b1) (b2) to [bend left] (b4) (t4)--(b5);
	\end{tikzpicture}};}
	& \tikz{
%	\node at (0,0) {};
	\node[TypeB, draw, line width=1.5pt, align=center] at (0,0) {
	\footnotesize Affine BMW \\
	\begin{tikzpicture}[xscale=.4, yscale=.4]
%		\node at (1,1.65){};
		\OverB[1,0][.1,.5]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,1);
			\draw[overB] (.3,-.1) -- (.3,1.1) (.5,-.1) -- (.5,1.1);
			\draw (.51,1.1) arc (0:360:3pt and 2pt);
			\draw (.51,-.1) arc (0:-180:3pt and 2pt);
		\OverB[.1,.5][1.75,1]
		\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x*.75+.25, 0){};	\node[V, TypeB] (t\x) at (\x*.75+.25, 1) {};}
	\draw[TypeB]  (b2) to [bend left=45] (b4) (t4)--(b5);
	\draw[overB] (t1)--(b3) (t3) to [bend right=45] (t5);
	\end{tikzpicture}};}
	& \tikz{
%	\node at (0,0) {};
	\node[TypeB, draw, line width=1.5pt, align=center] at (0,0) {
	\scriptsize 2-bdry BMW \\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
%		\node at (1,1.65){};
		\OverB[1,0][.1,.4]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,1);
			\draw[overB] (.3,-.1) -- (.3,1.1) (.5,-.1) -- (.5,1.1);
			\draw[TypeB] (.51,1.1) arc (0:360:3pt and 2pt);
			\draw[TypeB] (.51,-.1) arc (0:-180:3pt and 2pt);
		%RPOLE:
		\shade[left color=white,right color=white] (4.5,0) rectangle (4.7,1);
			\draw[overB] (4.7,-.1) -- (4.7,1.1) (4.5,-.1) -- (4.5,1.1);
			\draw[TypeB] (4.71,1.1) arc (0:360:3pt and 2pt);
			\draw[TypeB] (4.71,-.1) arc (0:-180:3pt and 2pt);
		\OverB[.1,.4][1.75,1]
		\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x*.75+.25, 0){};	\node[V, TypeB] (t\x) at (\x*.75+.25, 1) {};}
	\draw[TypeB]  (b2) to [bend left=45] (b4) (t4)--(b5);
	\draw[overB] (t1)--(b3) (t3) to [bend right=45] (t5);
	\end{tikzpicture}};}
%%%%%%%%%%%
\\
%%%%%%%%%%%
&\footnotesize Lie algs: & \tikz{
\node at (0,.65) {};
\node[TypeB, draw,  line width=1.5pt, fill=white, align=center] at (0,0) {
	\footnotesize Brauer algebra\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
	\node at (2,1.5){};
	\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x, 0){};	
		\node[V,TypeB] (t\x) at (\x, 1.25) {};}
	\draw (t1)--(b3) (t2)--(b1) (t3) to [bend right] (t5) (b2) to [bend left] (b4) (t4)--(b5);
	\end{tikzpicture}};}
& \tikz{
\node at (0,.65) {};
\node[TypeB, draw,  line width=1.5pt, fill=white, align=center] at (0,0) {
		\footnotesize Deg.\ aff.\  BMW\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
		\node at (2,1.25){};%spacers
		\foreach \x in {1, ..., 5}{
			\node[V, TypeB] (b\x) at (\x, 0){};	\node[V, TypeB] (t\x) at (\x, 1.25) {};}
		\draw (t1) to node[pos=.15, bV, fill=red]{}(b3)  (t3) to [bend right] node[pos=.3, bV, fill=red]{} (t5);
		\draw[] (t2)--(b1) (b2) to [bend left] (b4) (t4) to node[pos=.45, bV, fill=red]{} node[pos=.8, bV, fill=red]{} (b5);
	\end{tikzpicture}};}
&	\begin{tikzpicture}
	\node at (0,.65) {};
	\node[TypeB, draw, line width=1.5pt, align=center] at (0,0) {
		\footnotesize Deg.\ 2-bdry BMW\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
		\node at (2,1.25){};%spacers
		\foreach \x in {1, ..., 5}{
			\node[V, TypeB] (b\x) at (\x, 0){};	\node[V, TypeB] (t\x) at (\x, 1.25) {};}
		\draw (t1) to node[pos=.15, bV, fill=red]{}(b3)  (t3) to [bend right] node[pos=.3, C, blue, fill=white]{} (t5);
		\draw[] (t2)--(b1) (b2) to [bend left] (b4) (t4) to node[pos=.45, bV, fill=red]{} node[pos=.8, C, blue, fill=white]{} (b5);
	\end{tikzpicture}};
\end{tikzpicture}
%%%%%%%%%%%%%%%%%%%
\end{tabular}}~~~~~}

\smallskip\pause\small
{\bf Nazarov (95):} Introduced  \alert{degenerate affine Birman-Murakami-Wenzl (BMW) algebras}, built from Brauer algebras and their Jucys-Murphy elements. 

\smallskip\pause

{\bf H\"aring-Oldenburg (98) and Orellana-Ram (04):} Introduced the \alert{affine BMW algebras}. [OR04] gave the action on $M\otimes V^{\otimes k}$ commuting with the action of the quantum groups of types B, C, D.

\smallskip\pause

{\bf D.-Ram-Virk:} Used these centralizer relationships to study these two algebras simultaneously.  Results include \alert{computing the centers}, handling the \alert{parameters} associated to the algebras, computing powerful \alert{intertwiner operators}, etc. 



\medskip

{\color{white}\centerline{\begin{minipage}[t]{2.25in}
{\bf D.-Gonz\'alez-Schneider-Sutton:}\\
Constructing 2-boundary analogues \\(in progress.).
\end{minipage}~
\begin{minipage}[t]{2.25in}
{\bf Balagovic et al.:}\\
Signed versions and representations of periplectic Lie superalgebras.
\end{minipage}}}


\end{frame}

\begin{frame}
\hspace{-.5cm}\alert{Example: ``Admissibility conditions''}
\small
\centerline{\begin{tikzpicture}\draw[densely dotted, alert](0,0) to (11,0);\end{tikzpicture}}

\smallskip
\centerline{\begin{minipage}[]{2.1in}
\centerline{Affine BMW algebra}
\centerline{
$\begin{matrix}\begin{tikzpicture}[xscale=.5, yscale=.6]
		\Over[1,0][.1,.5]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,1);
			\draw[over] (.3,-.1) -- (.3,1.1) (.5,-.1) -- (.5,1.1);
			\draw[] (.51,1.1) arc (0:360:3pt and 2pt);
			\draw[] (.51,-.1) arc (0:-180:3pt and 2pt);
		\Over[.1,.5][1.75,1]
		\foreach \x in {1, ..., 5}{
		\node[V, ] (b\x) at (\x*.75+.25, 0){};	\node[V, ] (t\x) at (\x*.75+.25, 1) {};}
	\draw[over]  (b2) to [bend left=45] (b4) (t4)--(b5);
	\draw[over] (t1)--(b3) (t3) to [bend right=45] (t5);
	\end{tikzpicture}\end{matrix}$}
Closed loops:\\
\centerline{
$\begin{matrix}\begin{tikzpicture}[xscale=.5, yscale=.6]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,1.5);
			\draw[over] (.3,-.1) -- (.3,1.6) (.5,-.1) -- (.5,1.6);
			\draw[] (.51,1.6) arc (0:360:3pt and 2pt);
			\draw[] (.51,-.1) arc (0:-180:3pt and 2pt);
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 1.5) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw (1,.75) to [bend left=45]  (2,.75) to [bend left=45] (1,.75);
	\end{tikzpicture}\end{matrix},
	~~
\begin{matrix}\begin{tikzpicture}[xscale=.5, yscale=.6]
%		\node at (1,1.65){};
		\Over[1,.5][.1,1]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,2);
			\draw[over] (.3,-.1) -- (.3,2.1) (.5,-.1) -- (.5,2.1);
			\draw[] (.51,2.1) arc (0:360:3pt and 2pt);
			\draw[] (.51,-.1) arc (0:-180:3pt and 2pt);
		\Over[.1,1][1,1.5]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 2) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw[rounded corners=1] (1,1.5) to [bend left=35]  (2,1.5) to (2,.5) to [bend left=35] (1,.5);
	\end{tikzpicture}\end{matrix},~~
\begin{matrix}\begin{tikzpicture}[xscale=.5, yscale=.6]
%		\node at (1,1.65){};
		\Over[1,.5][.1,.875]
		\Over[1,1.25][.1,1.625]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,2.5);
			\draw[over] (.3,-.1) -- (.3,2.6) (.5,-.1) -- (.5,2.6);
			\draw[] (.51,2.6) arc (0:360:3pt and 2pt);
			\draw[] (.51,-.1) arc (0:-180:3pt and 2pt);
		\Over[.1,.875][1,1.25]
		\Over[.1,1.625][1,2]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 2.5) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw[rounded corners=1] (1,2) to [bend left=35]  (2,2) to (2,.5) to [bend left=35] (1,.5);
	\end{tikzpicture}\end{matrix}~~\cdots$}
\end{minipage}
%%%%%%%%%%
$\begin{matrix}\begin{tikzpicture}\draw[densely dotted, alert](0,0) to (0,3.5);\end{tikzpicture}\end{matrix}$~
\begin{minipage}[]{2.1in}
\centerline{Degenerate affine BMW algebra}
\centerline{
$\begin{matrix}\begin{tikzpicture}[xscale=.45, yscale=.5]
		\node at (2,1.25){};%spacers
		\foreach \x in {1, ..., 5}{
			\node[V, ] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 1.25) {};}
		\draw[] (t1) to (b3)  (t3) to [bend right]  (t5);
		\draw[] (t2) to node[pos=.75, bV, fill=red]{} (b1) (b2) to [bend left] (b4) (t4) to (b5);
	\node at (1,.1){}; 
	\end{tikzpicture}\end{matrix}$}
Closed loops:\\
\centerline{
$\begin{matrix}\begin{tikzpicture}[xscale=.5, yscale=.6]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 1.5) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw (1,.75) to [bend left=45]  (2,.75) to [bend left=45] (1,.75);
	\end{tikzpicture}\end{matrix},
	~~
\begin{matrix}\begin{tikzpicture}[xscale=.5, yscale=.6]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 2) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw[rounded corners=1] (1,1.25) to [bend left=35]  (2,1.25) 
							to (2,.75) to [bend left=35] (1,.75)
							to node[bV, fill=red]{} (1,1.25);
	\end{tikzpicture}\end{matrix},~~
\begin{matrix}\begin{tikzpicture}[xscale=.5, yscale=.6]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 2.5) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw[rounded corners=1] (1,1.75) to [bend left=35]  (2,1.75) 
							to (2,.75) to [bend left=35] (1,.75)
							to node[bV, fill=red, pos=.3]{}
								node[bV, fill=red, pos=.7]{} (1,1.75);
	\node at (1,2.5){}; \node at (1,-0){}; 
	\end{tikzpicture}\end{matrix}~~\cdots$}
\end{minipage}}
\centerline{\begin{tikzpicture}\draw[densely dotted, alert](0,0) to (11,0);\end{tikzpicture}}


\pause
The associated parameters of the algebra, e.g.\\
\centerline{
$\begin{matrix}\begin{tikzpicture}[scale=.4]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 1.5) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw (1,.75) to [bend left=45]  (2,.75) to [bend left=45] (1,.75);
	\end{tikzpicture}\end{matrix} = z_0\ \begin{matrix}\begin{tikzpicture}[scale=.4]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 1) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\end{tikzpicture}\end{matrix}\ ,
	\qquad
\begin{matrix}\begin{tikzpicture}[scale=.4]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 2) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw[rounded corners=1] (1,1.25) to [bend left=35]  (2,1.25) 
							to (2,.75) to [bend left=35] (1,.75)
							to node[bV, fill=red]{} (1,1.25);
	\end{tikzpicture}\end{matrix}=z_1\ \begin{matrix}\begin{tikzpicture}[scale=.4]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 1) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\end{tikzpicture}\end{matrix}\ ,\qquad
\begin{matrix}\begin{tikzpicture}[scale=.4]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 2.5) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\draw[rounded corners=1] (1,1.75) to [bend left=35]  (2,1.75) 
							to (2,.75) to [bend left=35] (1,.75)
							to node[bV, fill=red, pos=.3]{}
								node[bV, fill=red, pos=.7]{} (1,1.75);
	\node at (1,2.5){}; \node at (1,-0){}; 
	\end{tikzpicture}\end{matrix}=z_2\ \begin{matrix}\begin{tikzpicture}[scale=.4]
		\foreach \x in {1, 2}{
			\node[V] (b\x) at (\x, 0){};	\node[V, ] (t\x) at (\x, 1) {};}
	\draw[bend left=35] (b1) to  (b2) (t2) to (t1);
	\end{tikzpicture}\end{matrix}\ ,~~\cdots$}
aren't entirely free (more so in ``cyclotomic quotients''). \pause

\smallskip

\alert{Important insight:} As operators on tensor space $M \otimes V \otimes V$, 
\\ \smallskip
\centerline{$\displaystyle
\begin{matrix}\begin{tikzpicture}[scale=.4]
	\draw[rounded corners=1] (1,1.5) to node[bV, fill=red]{} (1,2) to [bend left=35]  (2,2) 
			 to (2,0) to [bend left=35] (1,0) to node[bV, fill=red]{} (1,.5);	
			 \draw[thick, dotted, alert] (1,1.5) to (1,.5);
	\draw[|-|] (.5,0) to node[red,fill=white]{\footnotesize$\ell$} (.5,2);
	\end{tikzpicture}\end{matrix} \in Z(U\fg) \otimes \CC \otimes \CC
	\qquad \text{ and } \qquad 
	\begin{matrix}\begin{tikzpicture}[scale=.4]
		\Over[1,0][.1,.375]
		\Over[1,1.75][.1,2.125]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,2.65);
			\draw[over] (.3,-.15) -- (.3,2.65) (.5,-.15) -- (.5,2.65);
			\draw[] (.51,2.65) arc (0:360:3pt and 2pt);
			\draw[] (.51,-.15) arc (0:-180:3pt and 2pt);
		\Over[.1,.375][1,.75]
		\Over[.1,2.125][1,2.5]
	\draw[rounded corners=1] (1,2.5) to [bend left=35]  (2,2.5) to (2,0) to [bend left=35] (1,0);
			 \draw[thick, dotted, alert] (1,1.5) to (1,.5);
	\draw[|-|] (-.5,0) to node[red,fill=white]{\footnotesize$\ell$} (-.5,2.5);
	\end{tikzpicture}\end{matrix} \in Z(U_q \fg) \otimes \CC \otimes \CC.
$} \smallskip
\hfill {\footnotesize ``Higher Casimir invariants''}
\end{frame}


\begin{frame}
\centerline{\hbox{\setlength{\tabcolsep}{3pt} 
\def\arraystretch{.7}\begin{tabular}{M@{\hspace{-.1in}}rMMMM@{\hspace{-.1in}}M}
\multirow{3}{*}{
      \begin{tikzpicture}
		\draw[white](0,0)  to 
			node[midway,black, TypeB] {\footnotesize \begin{tabular}{c}Orthogonal\\and\\symplectic\\
	\scriptsize(types B, C, D)\end{tabular}}  (0,3.75);
	\end{tikzpicture}
	}
 &&\footnotesize$V^{\otimes k}$ & \footnotesize$M \otimes V^{\otimes k}$ & \footnotesize$M \otimes V^{\otimes k} \otimes N$\\
%%%%%%%%%%%%%%%%%%%
&\footnotesize Qu.\ grps:& \tikz{
%	\node at (0,0) {};
	\node[TypeB, draw,  fill=white, line width=1.5pt,  align=center]  at (0,0) {
	\footnotesize BMW algebra\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
%	\node at (2,1.5){};
	\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x, 0){};	\node[V,TypeB] (t\x) at (\x, 1.25) {};}
	\draw (t1)--(b3)  (t3) to [bend right] (t5);
	\draw[overB]
			(t2)--(b1) (b2) to [bend left] (b4) (t4)--(b5);
	\end{tikzpicture}};}
	& \tikz{
%	\node at (0,0) {};
	\node[TypeB, draw, line width=1.5pt, align=center] at (0,0) {
	\footnotesize Affine BMW \\
	\begin{tikzpicture}[xscale=.4, yscale=.4]
%		\node at (1,1.65){};
		\OverB[1,0][.1,.5]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,1);
			\draw[overB] (.3,-.1) -- (.3,1.1) (.5,-.1) -- (.5,1.1);
			\draw (.51,1.1) arc (0:360:3pt and 2pt);
			\draw (.51,-.1) arc (0:-180:3pt and 2pt);
		\OverB[.1,.5][1.75,1]
		\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x*.75+.25, 0){};	\node[V, TypeB] (t\x) at (\x*.75+.25, 1) {};}
	\draw[TypeB]  (b2) to [bend left=45] (b4) (t4)--(b5);
	\draw[overB] (t1)--(b3) (t3) to [bend right=45] (t5);
	\end{tikzpicture}};}
	& \tikz{
%	\node at (0,0) {};
	\node[TypeB, draw, line width=1.5pt, align=center] at (0,0) {
	\scriptsize 2-bdry BMW \\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
%		\node at (1,1.65){};
		\OverB[1,0][.1,.4]
		%POLE:
		\shade[left color=white,right color=white] (.5,0) rectangle (.3,1);
			\draw[overB] (.3,-.1) -- (.3,1.1) (.5,-.1) -- (.5,1.1);
			\draw[TypeB] (.51,1.1) arc (0:360:3pt and 2pt);
			\draw[TypeB] (.51,-.1) arc (0:-180:3pt and 2pt);
		%RPOLE:
		\shade[left color=white,right color=white] (4.5,0) rectangle (4.7,1);
			\draw[overB] (4.7,-.1) -- (4.7,1.1) (4.5,-.1) -- (4.5,1.1);
			\draw[TypeB] (4.71,1.1) arc (0:360:3pt and 2pt);
			\draw[TypeB] (4.71,-.1) arc (0:-180:3pt and 2pt);
		\OverB[.1,.4][1.75,1]
		\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x*.75+.25, 0){};	\node[V, TypeB] (t\x) at (\x*.75+.25, 1) {};}
	\draw[TypeB]  (b2) to [bend left=45] (b4) (t4)--(b5);
	\draw[overB] (t1)--(b3) (t3) to [bend right=45] (t5);
	\end{tikzpicture}};}
%%%%%%%%%%%
\\
%%%%%%%%%%%
&\footnotesize Lie algs: & \tikz{
\node at (0,.65) {};
\node[TypeB, draw,  line width=1.5pt, fill=white, align=center] at (0,0) {
	\footnotesize Brauer algebra\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
	\node at (2,1.5){};
	\foreach \x in {1, ..., 5}{
		\node[V, TypeB] (b\x) at (\x, 0){};	
		\node[V,TypeB] (t\x) at (\x, 1.25) {};}
	\draw (t1)--(b3) (t2)--(b1) (t3) to [bend right] (t5) (b2) to [bend left] (b4) (t4)--(b5);
	\end{tikzpicture}};}
& \tikz{
\node at (0,.65) {};
\node[TypeB, draw,  line width=1.5pt, fill=white, align=center] at (0,0) {
		\footnotesize Deg.\ aff.\  BMW\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
		\node at (2,1.25){};%spacers
		\foreach \x in {1, ..., 5}{
			\node[V, TypeB] (b\x) at (\x, 0){};	\node[V, TypeB] (t\x) at (\x, 1.25) {};}
		\draw (t1) to node[pos=.15, bV, fill=red]{}(b3)  (t3) to [bend right] node[pos=.3, bV, fill=red]{} (t5);
		\draw[] (t2)--(b1) (b2) to [bend left] (b4) (t4) to node[pos=.45, bV, fill=red]{} node[pos=.8, bV, fill=red]{} (b5);
	\end{tikzpicture}};}
&	\begin{tikzpicture}
	\node at (0,.65) {};
	\node[TypeB, draw, line width=1.5pt, align=center] at (0,0) {
		\footnotesize Deg.\ 2-bdry BMW\\
	\begin{tikzpicture}[xscale=.4, yscale=.37]
		\node at (2,1.25){};%spacers
		\foreach \x in {1, ..., 5}{
			\node[V, TypeB] (b\x) at (\x, 0){};	\node[V, TypeB] (t\x) at (\x, 1.25) {};}
		\draw (t1) to node[pos=.15, bV, fill=red]{}(b3)  (t3) to [bend right] node[pos=.3, C, blue, fill=white]{} (t5);
		\draw[] (t2)--(b1) (b2) to [bend left] (b4) (t4) to node[pos=.45, bV, fill=red]{} node[pos=.8, C, blue, fill=white]{} (b5);
	\end{tikzpicture}};
\end{tikzpicture}
%%%%%%%%%%%%%%%%%%%
\end{tabular}}~~~~~}

\smallskip\small

{\bf Nazarov (95):} Introduced  \alert{degenerate affine Birman-Murakami-Wenzl (BMW) algebras}, built from Brauer algebras and their Jucys-Murphy elements. 

\smallskip

{\bf H\"aring-Oldenburg (98) and Orellana-Ram (04):} Introduced the \alert{affine BMW algebras}. [OR04] gave the action on $M\otimes V^{\otimes k}$ commuting with the action of the quantum groups of types B, C, D.

\smallskip

{\bf D.-Ram-Virk:} Used these centralizer relationships to study these two algebras simultaneously.  Results include \alert{computing the centers}, handling the \alert{parameters} associated to the algebras, computing powerful \alert{intertwiner operators}, etc. 



\medskip\pause

\centerline{\begin{minipage}[t]{2.25in}
{\bf D.-Gonz\'alez-Schneider-Sutton:}\\
Constructing 2-boundary analogues \\(in progress.).
\end{minipage}~\pause
\begin{minipage}[t]{2.25in}
{\bf Balagovic et al.:}\\
Signed versions and representations of periplectic Lie superalgebras.
\end{minipage}}

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


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





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

\medskip

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

\medskip

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

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





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

\medskip


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

\medskip


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

\medskip

\uncover<1->{\centerline{\includegraphics{diagrams-slide-FR}}
\centerline{\includegraphics{diagrams-slide-LR}}}

\end{frame}



\begin{frame}\small
\textbf{Theorem} (D.-Ram)
\begin{enumerate}
\item[\color{Univ}(1)]  {\sl Let $U=U_q \fg$ for any complex reductive Lie algebras $\fg$.\\ Let $M$, $N$, and $V$ be finite-dimensional modules. 

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

\item[\color{TypeA}(2)]  {\sl If $\fg = \fgl_n$, then (for correct choices of $M$, $N$, and $V$),  \\
the affine Hecke algebra of type $C$,  $H_k$, acts on 
$M \otimes (V)^{\otimes k} \otimes N$
and this action commutes with the action of $U$. }


\item[\color{TypeA2}(3)] {\sl If $\fg = \fgl_2$, then the action of the two-boundary Temperley-Lieb algebra factors through the T.L.\ quotient of $H_k$.}
\end{enumerate}
\pause

Some results:
\begin{enumerate}[(a)]
\item A diagrammatic intuition for $H_k$.
\item A combinatorial classification and construction of irreducible representations of $H_k$ (type C with distinct parameters) via central characters and generalizations of Young tableaux.
\item A classification of the representations of $TL_k$ in [dGN08] via central characters, including answers to open questions and conjectures regarding their irreducibility and isomorphism classes. 
\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}