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\def\SPACE{\phantom{\Big|}}
 \def\ANSWER[#1]{\uncover<2->{{\color{dblue} #1}}}
\title{The quasi-partition algebra}
\author[Zajj Daugherty]{Zajj Daugherty\\~\\
Joint with Rosa Orellana}
\institute{Dartmouth College and ICERM}
\date{March 19, 2013}
\begin{document}

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


\begin{frame}{Diagram algebras}
Everyone's favorite diagram algebra: \\
Group algebra of the symmetric group $S_k$

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



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

\end{enumerate}
\end{frame}

\begin{frame}{Modules {\small (vector spaces, with the group or algebra acting as matrices)}}


Simple $S_k$-modules are in bijection with partitions,  $\lambda \vdash k$

$$\lambda = \begin{picture}(40, 23)(0,2)
		\multiput(0,0)(10,0){4} {\line(0,1){20}} %V
		\multiput(0,0)(0,10){3} {\line(1,0){30}} %H
		%
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		\put(30,10) {\line(1,0){10}} %H
		%
		\put(40,20) {\line(0,-1){10}} %V
		%
		\put(0,-10) {\line(1,0){10}} %H
		%
		\put(0,0) {\line(0,-1){10}} %V
		\put(10,0) {\line(0,-1){10}} %V
		%
%		\only<3->{
%		\put(3,12){\text{\footnotesize\alert{0}}}
%		\put(1,2){\text{\footnotesize\alert{-1}}}
%		\put(1,-8){\text{\footnotesize\alert{-2}}}
%		%
%		\put(13,12){\text{\footnotesize\alert{1}}}
%		\put(13,2){\text{\footnotesize\alert{0}}}
%		%
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%		\put(23,2){\text{\footnotesize\alert{1}}}
%		%
%		\put(33,12){\text{\footnotesize\alert{3}}}
%		}
		\put(50,12){\small \color{alert2} 4}
		\put(50,2){\small \color{alert2} ~+3}
		\put(50,-8){\small \color{alert2} ~+1}
     \end{picture} 
$$
~
\centerline{\small (a collection of boxes piled up and to the left)}

\vfill

So, for example,
$$S^{\begin{tikzpicture}[scale=.2]
\draw (0,0)--(3,0); \draw (0,1)--(3,1);
\draw (0,0)--(0,1); \draw (1,0)--(1,1); \draw (2,0)--(2,1); \draw (3,0)--(3,1);
\end{tikzpicture}} \qquad 
S^{\begin{tikzpicture}[scale=.2]
\draw (0,0)--(1,0); \draw (0,1)--(2,1);\draw (0,2)--(2,2);
\draw (0,0)--(0,2); \draw (1,0)--(1,2); \draw (2,1)--(2,2); 
\end{tikzpicture}}\qquad \text{ and } \qquad
S^{\begin{tikzpicture}[scale=.2]
\draw (0,0)--(0,3); \draw (1,0)--(1,3);
\draw (0,0)--(1,0); \draw (0,1)--(1,1); \draw (0,2)--(1,2); \draw (0,3)--(1,3);
\end{tikzpicture}}
$$
are the simple $S_3$-modules (up to isomorphism). 
\end{frame}


\begin{frame}{Schur-Weyl duality and centralizer algebras}
{} {\bf Classical example:}  $S_k$ and $\GL_n$ have commuting actions on $\left( \CC^n\right)^{\otimes k}$, and their images fully centralize each in $\End\left(\left( \CC^n\right)^{\otimes k} \right)$.

\uncover<2->{\begin{block}{\bf Why this is exciting:}
		Centralizer relationship produces
			$$\left( \CC^n\right)^{\otimes k}  \cong \bigoplus_{\tiny \lambda \vdash k }  L(\lambda) \otimes S^\lambda
			\quad \text{ as a $\GL_n$-$S_k$ bimodule,}$$
where 		\begin{tabular}{rcl}
			$L(\lambda)$ &are distinct irreducible& $\GL_n$-modules\\
			$S^\lambda$ &are distinct irreducible& $S_k$-modules\\
			\end{tabular}
	\end{block}}
	
	\uncover<3->{
	For example, 
	\centerline{$\CC^n \otimes \CC^n \otimes \CC^n  = \left(L(\begin{minipage}{.6cm}\begin{tikzpicture}[scale=.2]
\draw (0,0)--(3,0); \draw (0,1)--(3,1);
\draw (0,0)--(0,1); \draw (1,0)--(1,1); \draw (2,0)--(2,1); \draw (3,0)--(3,1);
\end{tikzpicture}\end{minipage}) \otimes S^{\begin{tikzpicture}[scale=.2]
\draw (0,0)--(3,0); \draw (0,1)--(3,1);
\draw (0,0)--(0,1); \draw (1,0)--(1,1); \draw (2,0)--(2,1); \draw (3,0)--(3,1);
\end{tikzpicture}}\right) \oplus 
\left( L\big(\begin{minipage}{.41cm}\begin{tikzpicture}[scale=.2]
\draw (0,0)--(1,0); \draw (0,1)--(2,1);\draw (0,2)--(2,2);
\draw (0,0)--(0,2); \draw (1,0)--(1,2); \draw (2,1)--(2,2); 
\end{tikzpicture}\end{minipage}\big) \otimes S^{\begin{tikzpicture}[scale=.2]
\draw (0,0)--(1,0); \draw (0,1)--(2,1);\draw (0,2)--(2,2);
\draw (0,0)--(0,2); \draw (1,0)--(1,2); \draw (2,1)--(2,2); 
\end{tikzpicture}}\right) \oplus
\left( L\Big(\begin{minipage}{.21cm}\begin{tikzpicture}[scale=.2]
\draw (0,0)--(0,3); \draw (1,0)--(1,3);
\draw (0,0)--(1,0); \draw (0,1)--(1,1); \draw (0,2)--(1,2); \draw (0,3)--(1,3);
\end{tikzpicture}\end{minipage}\Big) \otimes
S^{\begin{tikzpicture}[scale=.2]
\draw (0,0)--(0,3); \draw (1,0)--(1,3);
\draw (0,0)--(1,0); \draw (0,1)--(1,1); \draw (0,2)--(1,2); \draw (0,3)--(1,3);
\end{tikzpicture}}\right)$}}
	
\end{frame}







\begin{frame}{Switching roles: the partition algebra}

Let $V$ be the permutation representation of $S_n$. 
$$\text{{\sl $n \times n$ matrices with 1's and 0's} \qquad i.e. \qquad $\sigma\cdot v_i = v_{\sigma(i)}$}$$
\vfill



Now let $S_n$ act diagonally on $V^{\otimes k}$: 
$$\sigma \cdot (v_{i_1} \otimes \cdots \otimes v_{i_k}) = v_{\sigma(i_1)} \otimes \cdots \otimes v_{\sigma(i_k)}$$

\pause 
\vfill

\alert{\bf What commutes? }\pause

Permutation of the factors again. \hfill \pause But lots more! \pause

$$\begin{tikzpicture}
\Nodes[0][4] \Nodes[1][4]
\draw (1,0)--(2,0)--(3,0)--(2,1)--(1,1)--(1,0);
\draw (3,1)--(4,1);
\draw[->] (5,.2)--(5,.8);
\node[right] at (5,.5) {$\varphi$};
\pause
\foreach \x in {1,...,3}{\node at (\x+.5, -.3) {\small $\otimes$};}
\node at (1,-.3) {$v_a$};
\node at (2,-.3) {$v_b$};
\node at (3,-.3) {$v_c$};
\node at (4,-.3) {$v_d$};
\pause
\node[left] at (.75,1.4) {$\delta_{a=b=c}$};
\node at (1.5,1.4) {$(v_a\otimes v_a)$};\pause
\node at (2.45, 1.4) { $\otimes$};
\node at (3.8,1.4) {$\big(\sum_{i=1}^n v_i \otimes v_i\big)$};
\end{tikzpicture}$$

\end{frame}



\begin{frame}{Set partitions}
Fix $k \in \ZZ_{>0}$, and let
$$[k] = \{1, \dots, k\} \qquad \text{ and } \qquad [k'] = \{1', \dots, k'\}.$$
\pause
We're interested in \alert{set partitions} of $[k] \cup [k']$.  \pause
Either as sets of sets
 $$d=\{ \{1,2,1'\}, \{3\}, \{2', 3', 4', 4\}\}$$
 or as diagrams (considering connected components) 
\begin{center}
	{
	\begin{tikzpicture}[scale=1]
		\foreach \x in {1,...,4} {
			\node[above] at (\x,1) {\tiny \x};
			\node[below] at (\x,0) {\tiny \x'};
			}
		\Nodes[0][4]	\Nodes[1][4]
		\draw (1,1)--(1,0)--(2,1) (2,0)--(3,0)--(4,0)--(4,1);
	\end{tikzpicture}\pause
	\qquad \VCenter{2}{$=$} \qquad 
	\begin{tikzpicture}[scale=1]
		\foreach \x in {1,...,4} {
			\node[above] at (\x,1) {\tiny \x};
			\node[below] at (\x,0) {\tiny \x'};
			}
		\Nodes[0][4]
		\Nodes[1][4]
		\draw (1,0)--(1,1)--(2,1) (4,1)--(3,0)--(4,0)--(4,1)--(2,0);
	\end{tikzpicture}
}\end{center}
{\small (Both encode the map $ \displaystyle v_a \otimes v_b \otimes v_c \otimes v_d \mapsto \delta_{b=c=d}(v_a \otimes v_a) \otimes \sum_{i=1}^n v_i \otimes v_b$)}
\end{frame}

\begin{frame}{The partition algebra}
Multiplying diagrams:
$$\begin{tikzpicture}[line width=1pt]
\uncover<3->{
	\foreach \x in {1,...,4}{
	\draw[dotted, dred] (\x,0)--(\x,-.5);
	}
}
%%%%%%%%%%%%%
\foreach \x in {1,...,4}{
 \filldraw [black] (\x,1) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
\node[above] at (\x, 1) {\scriptsize \color{dred} \x};
\uncover<1-2>{  \node[below] at (\x, 0) {\scriptsize \color{dred} \x'};}
}
\draw (1,1)--(2,1) (1,0)--(2,0) (3,0)--(4,0)--(4,1)--(3,0);
%%%%%%%%%%%%%
\pgftransformyshift{-1.6cm}
%%%%%%%%%%%%%
\uncover<2->{
	\foreach \x in {1,...,4}{
		 \filldraw [black] (\x,1) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
		  \node[below] at (\x, 0) {\scriptsize \color{dred} \x''};
	 }
	\draw  (3,1)-- (1,0) (4,1)--(4,0)--(3,0)--(2,0)--(4,1);
}
%%%%%%%%%%%%%
\pgftransformyshift{1cm}
\pgftransformxshift{5cm}
%%%%%%%%%%%%%
\uncover<4->{\node[left] at (.75,.5) {$= n^1$};
	\foreach \x in {1,...,4}{
		 \filldraw [black] (\x,1) circle (2pt);
		 \filldraw [black] (\x,0) circle (2pt);
		  \node[below] at (\x, 0) {\scriptsize \color{dred} \x'};
		  \node[above] at (\x, 1) {\scriptsize \color{dred} \x};
	 }
	\draw (1,1)--(2,1) (4,1)--(4,0)--(3,0)--(2,0)--(1,0)--(4,1);
}
\end{tikzpicture}$$
\pause\pause\pause\pause \vspace{-.25in}

The \alert{partition algebra} $P_k(n)$ is the $\CC$-span of the partition diagrams with this product.\\ 
\vfill
\pause

{\bf Nice facts:} \\\pause
($\alert{*}$) Associative algebra with identity $1=\{\{1,1'\}, \dots, \{k,k'\}\}$. \\\pause
($\alert{*}$) $\dim(P_k(n)) =$ the \alert{Bell number} $B(2k)$. \\\pause
($\alert{*}$) $S_n$ and $P_k(n)$ centralize each other in $\End(V^{\otimes k})$.
\end{frame}


\begin{frame}
Problem: $V$ is not irreducible! 
\begin{align*}
V &= \CC\{v_1, \dots, v_n\}\\
W & = \CC\{w_2, \dots, w_n\} & \text{ where } w_i = v_i - v_1,\\
T &  = \CC v, & \text{ where } v = v_1 + \cdots + v_n.
\end{align*}
Then $V = W \oplus T$.
\pause

\vfill

\centerline{\fbox{\bf \alert{The question:} What is $\End_{S_n}(W^{\otimes k})$?}}\pause
Equivalent: Which maps in $P_k(n)$ send $W^{\otimes k} \to W^{\otimes k}$?\pause

\vfill

A first hint: \\
\centerline{\begin{tikzpicture}[scale=.5]\node[left] at (1,.5){If $p = $};\Nodes[0][1] \Nodes[1][1]\node[right] at (1,.5){, \quad then $p \cdot v_i = v$. \qquad  So $p = n \pi_T$ projects onto $T$.};\end{tikzpicture}}\pause
\centerline{\begin{tikzpicture}[xscale=.4, yscale=.5]
	\node[left] at (-2,.5) {Let $p_i = $};
	 \foreach \x in {-2,0,1,2,4} {
	 \filldraw [black] (\x, 0) circle (2pt);
	 \filldraw [black] (\x, 1) circle (2pt);
	 }
	\draw  (0,1)--(0,0) (-2,1)--(-2,0) (2,1)--(2,0) (4,1)--(4,0);
	\node at (-1,.5) {$\dots$};
	\node at (3,.3) {$\dots$};
	\node at (1,1.5) {$i$};
	\node[right] at (5,.5){Then $p_i: V ^{\otimes k} \to V^{\otimes i-1}  \otimes T \otimes V^{\otimes k-i}$.};
	\end{tikzpicture}}\pause
Any diagram $d$ an isolated vertex satisfies $d = p_i d'$ or $d = d' p_i$. 
	\end{frame}

\begin{frame}
Let $d$ be a  diagram on $2k$ vertices. \\
\textbf{Two maps in $\End(W^{\otimes k})$:}\pause
\begin{enumerate}
\item  Put $d \in P_k(n)$, and project its image onto $W^{\otimes k}$:
$$\bar{d} : W^{\otimes k} \xrightarrow{d} V^{\otimes k} \xrightarrow{\pi_W^{\otimes k}} W^{\otimes k}.$$\pause
\alert{Good:} Commutes with $S_n$ \\
\alert{Bad:} What are they, and how do we multiply? \pause
\item Let 
$f: \{v_1, \dots, v_{n-1} \} \to \{w_2, \dots, w_n\} \qquad v_i \mapsto w_{i+1}.$\\
Put $d \in P_k(n-1)$, and consider 
$$[d] : W^{\otimes k} \xrightarrow{f^{-1}} V_{n-1}^{\otimes k} \xrightarrow{d}  V_{n-1}^{\otimes k}\xrightarrow{f} W^{\otimes k}$$\pause
\alert{Good:} We know what they are and how to multiply them.\\
\alert{Bad:} They don't commute with $S_n$. 
\end{enumerate}
\pause
\vfill
Goal: Express $\bar{d}$ in terms of $[d']$'s.

\end{frame}



\begin{frame}
If $X$ is a set of vertices, the \alert{isolation} of $d$ (at $X$) is 
$d_X$ , { the diagram constructed from $d$ by isolating all vertices in $X$.}

\vfill

For example, if $X=\{ 1', 4'\}$ and 
$$\begin{tikzpicture}[scale=.75]
	\Nodes[1][4];
	\Nodes[0][4];
	\node at (0,.5) {$d =$};
	\draw (1,0)--(1,1)--(2,0)--(1,0) ;
	\draw (2,1)--(3,1)--(4,1) (4,0)--(3,0) ;
	\foreach \x in {1, ..., 4}{
	\node[above] at (\x, 1) {\tiny $\x$};
	\node[below] at (\x, 0) {\tiny $\x'$};
	}
	\end{tikzpicture}
\qquad \VCenter{1.5}{then} \qquad 
\begin{tikzpicture}[scale=.75]
	\Nodes[1][4];
	\Nodes[0][4];
	\node at (0,.5) {$d_X =$};
	\draw (1,1)--(2,0)  (1,0);
	\draw (2,1)--(3,1)--(4,1); %(4,0)--(3,0)  ;
	\foreach \x in {1, ..., 4}{
	\node[above] at (\x, 1) {\tiny $\x$};
	\node[below] at (\x, 0) {\tiny $\x'$};
	}
	\end{tikzpicture}	
	$$
We can also place an order on diagrams, where $d' \leq d$ if $d'$ is a refinement of $d$. In particular, $d_X \leq d$. 
\end{frame}


\begin{frame}
Define the \alert{quasi-partition algebra} as  $QP_k(n) = \End_{S_n} (W^{\otimes k})$. \\
Let $\cD = \{$ diagrams $d$ without isolated verticies $\}$.  \pause

\begin{theorem}[D.-Orellana]
If $d \in \cD$ then 
$$\bar d = [d] + \sum_{X \subseteq [k] \cup [k']} c_X [d_X],$$
where $c_X$ is a (totally explicit) rational function in $1/n$.
\end{theorem}\pause
\only<3-4>{For example,
 $$
 \begin{tikzpicture}[scale=.75]
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0) (1,1)--(2,1);
\draw (1,1.2)--(2,1.2);
\ShiftX{1in}
\node[left] at (.5,.5) {$=$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0) (1,1)--(2,1);
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \ShiftX{1in}
\node[left] at (.75,.5) {$+$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,1)--(2,1);
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \ShiftX{1in}
\node[left] at (.75,.5) {$-\frac{1}{n}$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0) ;
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \ShiftX{1in}
\node[left] at (.75,.5) {$-\frac{1}{n}$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \end{tikzpicture}
 $$\pause
 $$
 \begin{tikzpicture}[scale=.75]
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0)--(2,1)--(1,1)--(1,0);
\draw (1,1.2)--(2,1.2);
\ShiftX{1in}
\node[left] at (.5,.5) {$=$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0)--(2,1)--(1,1)--(1,0);
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \ShiftX{1in}
\node[left] at (.75,.5) {$-\frac{1}{n}$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0)--(1,1)--(1,0);
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \ShiftX{1in}
\node[left] at (.75,.5) {$-\frac{1}{n}$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0)--(2,1)--(1,0);
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \ShiftX{1in}
\node[left] at (.85,.5) {$+\frac{1}{n^2}$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,0)--(2,0);
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
\ShiftX{1in}
\node[left] at (.85,.5) {$+\frac{1}{n^2}$};
 \Nodes[0][2] \Nodes[1][2]
 \draw (1,1.2)--(.8,1.2)--(.8,-.2)--(1,-.2);
 \draw (2,1.2)--(2.2,1.2)--(2.2,-.2)--(2,-.2);
 \end{tikzpicture}
 $$}
 \pause  \pause
 \begin{cor}
 $QP_k(n)$ has basis $\{\bar d ~|~ d \in \cD\}$, and thus has dimension  \[\sum\limits_{j=1}^{2k} (-1)^{j-1}B(2k-j)+1, \qquad \text{ where $B(r)$ is the Bell number.}\]
 \end{cor}\vspace{-.25in}
 \pause
 \begin{cor}
 If $d_1, d_2 \in \cD$, 
$$ \bar d_1 \bar d_2 = \sum_{d \leq d_1 d_2} c_d \bar d.$$
In particular, if $d_1 d_2 \notin \cD$, then  $\bar d_1 \bar d_2=0$.
 
 \end{cor}
 
 
 
\end{frame}


\begin{frame}
So \emph{functionally}, $QP_k(n)$ is a subalgebra of $P_k(n-1)$. 


It's generated by projections of 
$$\begin{tikzpicture}[xscale=.4, yscale=.5]
	\node[left] at (-2,.5) {$b_i = $};
	 \foreach \x in {-2,0,1,...,3,5} {
	 \filldraw [black] (\x, 0) circle (2pt);
	 \filldraw [black] (\x, 1) circle (2pt);
	 }
	\draw (1,0)--(1,1)--(2,1)-- (2,0)--(1,0);
	\draw  (0,1)--(0,0) (-2,1)--(-2,0) (3,1)--(3,0) (5,1)--(5,0);
	\node at (-1,.5) {$\dots$};
	\node at (4,.4) {$\dots$};
	\node at (1,1.5) {$i$};
	\end{tikzpicture} 
\qquad	
\begin{tikzpicture}[xscale=.4, yscale=.5]
	\node[left] at (-2,.5) {$s_i = $};
	 \foreach \x in {-2,0,1,...,3,5} {
	 \filldraw [black] (\x, 0) circle (2pt);
	 \filldraw [black] (\x, 1) circle (2pt);
	 }
	\draw (1,0)--(2,1)  (1,1)--(2,0);
	\draw  (0,1)--(0,0) (-2,1)--(-2,0) (3,1)--(3,0) (5,1)--(5,0);
	\node at (-1,.5) {$\dots$};
	\node at (4,.4) {$\dots$};
	\node at (1,1.5) {$i$};
	\end{tikzpicture} 
$$
$$
\begin{tikzpicture}[xscale=.4, yscale=.5]
	\node[left] at (-2,.5) {$e_i =$};
	 \foreach \x in {-2,0,1,...,3,5} {
	 \filldraw [black] (\x, 0) circle (2pt);
	 \filldraw [black] (\x, 1) circle (2pt);
	 }
	%\Nodes[0][4];
	\draw (1,0) (1,1)--(2,1) (2,0)--(1,0);
	%\draw (2,0)--(3,0)--(3,1)--(2,0) ;
	\draw  (0,1)--(0,0)  (-2,1)--(-2,0) (3,1)--(3,0) (5,1)--(5,0);
	\node at (-1,.5) {$\dots$};
	\node at (4,.4) {$\dots$};
	\node at (1,1.5) {$i$};
	\end{tikzpicture} 
\qquad\begin{tikzpicture}[yscale=.5, xscale=.4]
	\node[left] at (-2,.5) {$t_i =  $};
	 \foreach \x in {-2,0,1,...,4,6} {
	 \filldraw [black] (\x, 0) circle (2pt);
	 \filldraw [black] (\x, 1) circle (2pt);
	 }
	\Nodes[0][4];
	\draw (1,0)--(1,1)--(2,1)-- (1,0);
	\draw (2,0)--(3,0)--(3,1)--(2,0) ;
	\draw (4,0)--(4,1) (0,1)--(0,0) (-2,1)--(-2,0) (6,1)--(6,0);
	\node at (-1,.5) {$\dots$};
	\node at (5,.5) {$\dots$};
	\node at (1,1.5) {$i$};\end{tikzpicture}$$\pause
With relations that look like 
$$
\begin{array}{c@{\quad}|@{\quad} c} 
\text{in }P_k(n-1): & \text{in }QP_k(n):\\\hline~\\
s_i^2 = 1 & \bar s_i^2 = 1\\~\\
s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}& \bar s_i \bar s_{i+1} \bar s_i = \bar s_{i+1}\bar s_i \bar s_{i+1}\\~\\
e_i^2 = (n-1) e_i & \bar e_i^2 = (n-1) \bar e_i\\~\\
b_i^2 = b_i & \bar b_i^2 =  \hbox{$\frac{n-2}{n}$}\bar b_i  + \hbox{$\frac{1}{n^2}$}\bar e_i
\end{array}
$$

\end{frame}





\begin{frame}{Representation theory}
Tensoring rule for $W = S^{(n-1,1)}$
$$S^{\lambda} \otimes W = c(\lambda)S^{\lambda}\oplus \bigoplus_{\mu \in \Lambda} S^\mu$$
where $\Lambda$ is the set of partitions gotten from $\lambda$ by moving any corner box to another place, and $c(\lambda) = \#$ corner boxes $- 1$. \pause

Example: 
$${\tiny \yng(5,2)} \otimes {\tiny \yng(6,1)} = {\tiny \yng(5,2) }\oplus {\tiny \yng(6,1)} 
		\oplus {\tiny \yng(4,3)} \oplus {\tiny \yng(5,1,1)} \oplus {\tiny \yng(4,2,1)}$$

\pause

Assume $n>>1$. We can forget the top row: 
$$\overline{\tiny \yng(2)} \quad \otimes \quad\overline{\tiny \yng(1)} \quad=\quad \overline{\tiny \yng(2) }\quad\oplus\quad \overline{\tiny \yng(1)} 
		\quad\oplus\quad \overline{\tiny \yng(3)} \quad\oplus\quad \overline{\tiny \yng(1,1)} \quad\oplus\quad \overline{\tiny \yng(2,1)}$$




\end{frame}

\begin{frame}{Bratteli diagram for $QP_k(n)$}
\xymatrix{
\tiny{\yng(1)} 
\only<2->{		
		\ar@{-}[d]
		\ar@{-}@/^.1pc/[dr]
		\ar@{-}@/^.2pc/[drr]
		\ar@{-}@/^.5pc/[drrr]}
\\
\uncover<2->{\emptyset }
\only<3->{		\ar@{-}@/^.1pc/[dr]}
& \uncover<2->{\tiny{\yng(1)} }
\only<3->{		\ar@{-}@/_.1pc/[dl]
		\ar@{-}[d]
		\ar@{-}@/^.1pc/[dr]
		\ar@{-}@/^.1pc/[drr]}
&\uncover<2->{ \tiny{\yng(2)} }
\only<3->{		\ar@{-}@/_.1pc/[dl]
		\ar@{-}[d]
		\ar@{-}@/^.1pc/[dr]
		\ar@{-}@/^.1pc/[drr]
		\ar@{-}@/^.1pc/[drrr]}
&\uncover<2->{ \tiny{\yng(1,1)}}
\only<3->{		\ar@{-}@/_.1pc/[dll]
		\ar@{-}@/_.1pc/[dl]
		\ar@{-}[d]
		\ar@{-}@/^.1pc/[drr]
		\ar@{-}@/^.5pc/[drrr]}\\
\uncover<3->{	\emptyset} & \uncover<3->{	\tiny{\yng(1)}}
 & \uncover<3->{	\tiny{\yng(2)}}
  &\uncover<3->{	 \tiny{\yng(1,1)}} & 
  \uncover<3->{	\tiny{\yng(3)}} & 
  \uncover<3->{	\tiny{\yng(2,1)} }
\only<4->{			\ar@{-}@/_.5pc/[dlll]
		\ar@{-}@/_.3pc/[dll]
		\ar@2{-}@/_.1pc/[dl]
		\ar@{-}[d]
		\ar@{-}@/^.1pc/[dr]
		\ar@{-}@/^.1pc/[drr]}
&\uncover<3->{	 \tiny{\yng(1,1,1)}}\\
&\uncover<4->{	\cdots}&\uncover<4->{\tiny{\yng(2)}} & \uncover<4->{\tiny{\yng(1,1)}} & \uncover<4->{\tiny{\yng(2,1)} }& \uncover<4->{\tiny{\yng(3,1)} }& \uncover<4->{\tiny{\yng(2,2)}} & \uncover<4->{\tiny{\yng(2,1,1)}} &\uncover<4->{\cdots}
}
\vfill

\uncover<5->{
(Dimensions expressed explicitly in terms of numbers of standard tableaux  and Bell numbers.)}

\end{frame}

\end{document}