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 \def\ANSWER[#1]{\uncover<2->{{\color{dblue} #1}}}
\title{Centralizers of the infinite symmetric group}
\author[Zajj Daugherty]{Zajj Daugherty\\~\\
Joint with Peter Herbrich}
\institute{Dartmouth College}
\date{November 26, 2013}
\begin{document}

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


\begin{frame}{Schur-Weyl duality -- a warm-up}
Start with the symmetric group $S_k$: permutations of $1, \dots, k$. \\
Depict using permutation diagrams:


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

%\begin{frame}{Schur-Weyl duality -- a warm-up}
%The representation theory is also combinatorial: 
%
%Simple $S_k$-modules are in bijection with partitions,  $\lambda \vdash k$
%
%\centerline{ \begin{tikzpicture}[xscale=.4, yscale=-.4]
%	\node[left] at (-.6,1.5){$\lambda =$};
%		\Part{4,3,1}
%		\node[right, structure] at (5,.5) {4};
%		\node[right, structure] at (5,1.5) {+3};
%		\node[right, structure] at (5,2.5) {+1};
%		\end{tikzpicture}}~
%\centerline{\small (a collection of boxes piled up and to the left)}
%
%\vfill\pause
%
%So, for example,
%$$S^{\mPART{3}} \qquad 
%S^{\mPART{2,1}}
%	\qquad \text{ and } \qquad
%S^{\mPART{1,1,1}}
%$$
%are the simple $S_3$-modules (up to isomorphism). 
%\end{frame}


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

\uncover<2->{\begin{block}{Why this is exciting: Huge transfer of information!}
		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{$\CC^n \otimes \CC^n \otimes \CC^n  \cong \left(
		G^{\begin{tikzpicture}[xscale=.2, yscale=-.2]
			\Part{3}\end{tikzpicture}} \otimes 
		S^{\begin{tikzpicture}[xscale=.2, yscale=-.2]
			\Part{3}\end{tikzpicture}}\right) \oplus 
	\left( 		G^{\begin{tikzpicture}[xscale=.2, yscale=-.2]
			\Part{2,1}\end{tikzpicture}} \otimes 
		S^{\begin{tikzpicture}[xscale=.2, yscale=-.2]
			\Part{2,1}\end{tikzpicture}}\right) \oplus
	\left(		G^{\begin{tikzpicture}[xscale=.2, yscale=-.2]
			\Part{1,1,1}\end{tikzpicture}} \otimes 
		S^{\begin{tikzpicture}[xscale=.2, yscale=-.2]
			\Part{1,1,1}\end{tikzpicture}}\right)$}}
	
\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{What commutes? }\pause

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

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

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

\begin{frame}{A connection to symmetric functions}
As a consequence of the commuting relationship,
the $S_n$-invariants in $V^{\otimes k}$ form a natural $P_k(n)$-module. \pause
In fact, a basis for the $S_n$-invariants is indexed by set partitions of $[k]$, i.e. half diagrams:\pause

{\small 
\begin{align*}
\begin{matrix}\begin{tikzpicture}[line width=1pt, scale=.5]
\Nodes[0][3]
\end{tikzpicture}
\end{matrix} &\quad
\longleftrightarrow\quad
m_{\{\{1\}, \{2\}, \{3\} \}} 
&= \sum_{1 \leq a,b,c \leq n} v_a \otimes v_b \otimes v_c \\
%&& \longleftrightarrow~ \sum_{1 \leq a,b,c \leq n} x_a x_b x_c 
\\
\begin{matrix}\begin{tikzpicture}[line width=1pt, scale=.5]
\Nodes[0][3]
\draw[bend right] (1,0) to(3,0);
\end{tikzpicture}
\end{matrix} &\quad
\longleftrightarrow\quad
m_{\{\{1,3\}, \{2\} \}} 
&= \sum_{1 \leq a,b \leq n} v_a \otimes v_b \otimes v_a~
%&& \longleftrightarrow~ \sum_{1 \leq a,b \leq n} x_a x_b x_a 
\end{align*}\pause}
And the action is still by concatenation:


$$\begin{tikzpicture}[line width=1pt, scale=.75]
\uncover<5->{
	\foreach \x in {1,...,4}{
	\draw[very thick, dotted, dred] (\x,0)--(\x,-.65);
	}
}
%%%%%%%%%%%%%
\draw[dotted, gray] (.8,-.1)--(.8,1.2)--(4.2,1.2)--(4.2,-.2)--(.8,-.2);
\foreach \x in {1,...,4}{
 \filldraw [black] (\x,1) circle (2pt);
 \filldraw [black] (\x,0) circle (2pt);
}
\draw (1,1)--(2,1)  (3,0)--(4,0)--(4,1)--(3,0);
	\node[left] at (.75,.5){\small $d:$};
%%%%%%%%%%%%%
\pgftransformyshift{-1.75cm}
%%%%%%%%%%%%%
	\draw[dotted, gray] (.8,.8)--(.8,1.2)--(4.2,1.2)--(4.2,.8)--(.8,.8);
	\foreach \x in {1,...,4}{
		 \filldraw [black] (\x,1) circle (2pt);}
	\draw  (3,1)--(4,1);
	\node[below] at (2.5, .9){\small$m_{\{\{1\},\{2\},\{3,4\} \}}$};
%%%%%%%%%%%%%
\pgftransformyshift{.8cm}
\pgftransformxshift{5cm}
%%%%%%%%%%%%%
\uncover<5->{\node[left] at (.75,1) {\small$= n^2$};
	\foreach \x in {1,...,4}{
		 \filldraw [black] (\x,1) circle (2pt);
	 }
	\draw (1,1)--(2,1);
	\draw[dotted, gray] (.8,.8)--(.8,1.2)--(4.2,1.2)--(4.2,.8)--(.8,.8);
}
\uncover<5->{\node[below] at (2.5,.9) {\small$m_{\{\{1,2\},\{3\},\{4\}\}}$};}
\end{tikzpicture}$$

\end{frame}

\begin{frame}{A connection to symmetric functions}

Identify  $V^{\otimes k}$ with the homogeneous degree-$k$ elements of $\CC [x_1, \dots, x_n]$ with non-commuting variables, via $v_a \leftrightarrow x_a$:\pause
$$
v_a \otimes v_b \otimes v_a \quad \longleftrightarrow\quad x_ax_bx_a \quad \text{ and } \quad 
v_a \otimes v_b \otimes v_b  \quad \longleftrightarrow\quad  x_ax_b^2.    
$$\pause
So $P_k(n)$ acts on the degree $k$ homogeneous elements of $\CC[\mathbf{x}]^{S_n}$. \pause

\medskip

\alert{\bf Hopf algebras and symmetric functions: }\\\pause
$(\alert{*})$ Gessel (1984): duality between the Solomon descent algebra and the Hopf algebra of the quasi-symmetric functions. \\\pause
$(\alert{*})$ Malvenuto-Reutenauer (1995) use Schur-Weyl duality to connect the graded Hopf algebra $\bigoplus_k \CC S_k$ to other classes of symmetric functions.\\\pause
$(\alert{*})$ Aguiar-Orellana (2008) generalize [MR95] to connect a bigger diagram Hopf algebra  (of \emph{uniform block permutations}) and symmetric functions in non-commuting variables. \pause

\medskip

\alert{Issue:} finitely many versus countably many variables! 

\end{frame} 



\begin{frame}{Moving to the infinite symmetric group}

Natural inclusion: $S_n \subset S_{n+1}$ as permutations fixing $n+1$. \\
Consider the limit 
$$S_1 \hookrightarrow S_2 \hookrightarrow S_3 \hookrightarrow \cdots \to S_\infty,$$
so that $S_\infty$ is the group of bijections on $\NN$ which fix all but finitely many elements. \\\pause
Let $S_\infty$ act on the set $\{v_i\}_{i \in \NN}$ by $\sigma \cdot v_i = v_{\sigma(i)}.$ 


\bigskip\pause

\alert{\bf Want:} \\
\begin{enumerate}[(1)]
\item A vector space $V$ containing a countable linearly independent
subset $\{v_{i}\}_{i\in\NN}$;
\item an appropriate notion of $V^{\otimes k}$; and 
\item an algebra of endomorphisms on $V^{\otimes k}$  
\begin{enumerate}[i.] 
\item whose elements are determined by their image on $v_i$'s, and 
\item which contains $S_\infty$ via the above action. 
\end{enumerate}
\end{enumerate}

\end{frame}




\begin{frame}
Three examples explored:
\begin{enumerate}
\item Countable dimensional vector space $V = \CC^{(\NN)} = \CC\{v_1, v_2, ...\}$. \\ {\small
\uncover<2->{\alert{Good:} Well-behaved vector space. \\
\alert{Bad:} No non-trivial $S_\infty$ invariants! \hfill e.g.\ if $k=1,  \sum_i v_i  \notin V$)\\
\hfill (Sam-Snowden 2013: representation theoretic stability)}}

\smallskip

\item Banach space of $p$-power summable sequences 
$$V = \{ v = (a_1,a_2, \dots) \in \CC^\NN ~|~ ||v||_p < \infty\}.$${\small
\uncover<3->{\alert{Good:} Can get all the necessary $S_\infty$ invariants in each degree.\\
\alert{Bad:} Must restrict to bounded maps, yielding restrictive results.} }


\smallskip

\item Banach space of $\ell^\infty$ bounded sequences 
$$V = \{ v = (a_1,a_2, \dots) \in \CC^\NN ~|~ ||v||_\infty < \infty\}.$${\small
\uncover<4->{\alert{Good:} Has all the $S_\infty$ invariants, and admits a larger set of maps.\\
\alert{Bad:} Even bounded maps aren't well-behaved for our purposes.}}
\end{enumerate}



\uncover<5->{\small
\alert{Good/Bad?} \\
2 and 3 yield non-unitary and non-semisimple representations!}
\end{frame}


\begin{frame}{1. Countable dimensional vector space $V = \CC^{(\NN)} $}

If the $\varphi \in \End(V^{\otimes k})$ commutes with the action of $S_\infty$, it acts like a linear combination of partition algebra diagrams. 

\medskip

Additionally, to be in $\End(V^{\otimes k})$, its image must be a finite linear combination of $v_i$'s.
\pause

$$\begin{tikzpicture}[line width=1pt]
\Nodes[0][2] \Nodes[1][2]
\draw (1,0)--(1,1)--(2,1)--(1,0);
\uncover<3->{
\draw[|->] (.5,.2)--(.5,.8);
\node[left] at (.5,.5) {$\varphi$};
\foreach \x in {1}{\node at (\x+.5, -.3) {\small $\otimes$};}
\node at (1,-.3) {$v_a$};
\node at (2,-.3) {$v_b$};
\node at (1,1.4) {$v_a$};
\node at (1.5, 1.4) { $\otimes$};
\node at (2,1.4) {$v_a$};}
\uncover<4->{\node at (1.5,-1) {Yes!};}
\end{tikzpicture}
\qquad
\begin{tikzpicture}[line width=1pt]
\Nodes[0][2] \Nodes[1][2]
\draw (1,0)--(2,0)--(1,1)--(1,0);
\uncover<3->{
\draw[|->] (.5,.2)--(.5,.8);
\node[left] at (.5,.5) {$\varphi$};
\foreach \x in {1}{\node at (\x+.5, -.3) {\small $\otimes$};}
\node at (1,-.3) {$v_a$};
\node at (2,-.3) {$v_b$};
\node[left] at (.85,1.4) {$\delta_{a,b}\Big($};
\node at (1,1.4) {$v_a$};
\node at (1.45, 1.4) { $\otimes$};
\node[right] at (1.5,1.4) {$\big(\sum_{i} v_i\big)\Big)$};}
\uncover<4->{\node at (1.5,-1) {No!};}
\end{tikzpicture}
$$

\pause\pause

\medskip\pause

\alert{Result}: The \emph{top-propagating partition algebra}, generated by diagrams with no block isolated to the top. \\
(Sam-Snowden: the \emph{upward partition category} glues all $k$ together)


\end{frame}


\begin{frame}{2. Banach space of $p$-power summable sequences }
Place a metric $\mu$ on $\CC^\NN$ so that \\
\centerline{$\big|\big| \sum_i v_i \big|\big|_p = \left|\left| (1,1,1, \dots) \right|\right|_p = \sum_i \mu_i^p <\infty.$}
{\footnotesize (Enough to get all expected invariants in the closure of $V^{\otimes k}$ for each $k$.)}\pause

We restrict to continuous/bounded endomorphisms $\cB( \overline{V^{\otimes k}})$. \pause

\medskip 

Again, if the $\varphi \in \cB(\overline{V^{\otimes k}})$ commutes with $S_\infty$, it acts like a linear combination of partition algebra diagrams. \pause

However, boundedness then additionally restricts to maps whose image on simple tensors is a permutation of factors.\\
 {\footnotesize(All other  partition diagrams have unbounded images). }

\medskip\pause

\alert{Result}: The algebra of \emph{uniform block permutations}, generated by diagrams whose blocks have the same size on top as on bottom. 

\def\UNIT{.65}
$${
\begin{matrix}
 \begin{tikzpicture}[scale=\UNIT, line width=1pt ]
	\Nodes[1][4] \Nodes[0][4]
  \draw (1,1)--(1,0) to [bend left] (3,0)--(2,1)--(1,1) (3,1)--(2,0) (4,0)--(4,1); 	
\node[left] at (0,.5) {Yes:};
 \end{tikzpicture}
\end{matrix}\qquad\qquad\qquad
\begin{matrix}
 \begin{tikzpicture}[scale=\UNIT, line width=1pt ]
	\Nodes[1][4] \Nodes[0][4]
   \draw (1,1)--(1,0)--(2,1)--(1,1) (2,0)--(3,0)--(4,0)--(4,1)--(3,1)--(2,0); 	
\node[left] at (0,.5) {No:};
 \end{tikzpicture}
\end{matrix} 
}$$

\hfill {\small (Same algebra as in Aguiar-Orellana!)}
\end{frame}

\def\Mat{\mathrm{Mat}}

\begin{frame}{3. Banach space of $\ell^\infty$-bounded sequences}
Sequences $(a_1, a_2, \dots) \in \CC^{\NN}$ whose entries are bounded. 

\medskip

\alert{Issue:} Even $\ell^\infty$-bounded endomorphisms are not determined by their images on $\{ v_i \}_{i \in \NN}.$ \pause So let 
$\cB_\Mat( \overline{V^{\otimes k}})$ be the algebra of $\ell^\infty$-bounded maps which can be written as matrices. \\
\hfill{\footnotesize (The sums across rows are $\ell_\infty$ bounded.)} 

\pause
\bigskip

\alert{Result}: The \emph{bottom-propagating partition algebra}, generated by diagrams with no block isolated to the bottom. \hfill
{\footnotesize (Isomorphic to case 1)}

\bigskip

$$\begin{tikzpicture}[line width=1pt]
\Nodes[0][2] \Nodes[1][2]
\draw (1,0)--(1,1)--(2,0)--(1,0);
\node[left] at (.5,.5) {Yes:};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[line width=1pt]
\Nodes[0][2] \Nodes[1][2]
\draw (1,0)--(1,1)--(2,1)--(1,0);
\node[left] at (.5,.5) {No:};
\end{tikzpicture}$$

\end{frame}


\begin{frame}{Putting it back into context}

\alert{Remark 1}: Orellana et al.\ (in progress) show that if a diagram Hopf algebra (as in [MR95] or [AO08]) is built from partition diagrams, those diagrams can have no blocks isolated to the top or bottom rows.  

\smallskip

Case 1: no application to symmetric functions.\\
Case 2: tied to symmetric functions in [AO08].\\
Question: Is there a fix for case 3?

\pause
\bigskip

\alert{Remark 2:} For all three cases, even for $k=1$, the centralizer algebra is spanned by $\begin{tikzpicture}[scale = .55] \filldraw [black] (1,.5) circle (2pt); \filldraw [black] (1,0) circle (2pt);\draw(1,.5)--(1,0);\end{tikzpicture}$~\!, so is isomorphic to $\CC$.\\ However, in cases 2 and 3, we expected more since $V$ has an invariant subspace. This discrepancy comes from the fact that the action of $S_\infty$ is not semisimple. 

\smallskip

Question: Can we use this framework to study certain non-unitary representations of $S_\infty$?

\end{frame}


\end{document}











