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

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

\definecolor{dred}{rgb}{.7,0,.1}
\definecolor{dblue}{rgb}{.3,0,.8}
\definecolor{blueish}{rgb}{0.5, 0, .75}
\definecolor{dgreen}{rgb}{0,.6,.1}
\definecolor{dorange}{rgb}{.9,.3,0}
\definecolor{purple}{rgb}{.7,.1,1}
\definecolor{dgrey}{rgb}{.4,.4,.4}

 \newcommand{\BLUE}[1]{{\color{blue}#1}}
 \newcommand{\NOTE}[1]{{\color{red}#1}}

\usecolortheme[rgb={0,0,1}]{structure}
\setbeamercolor{alerted text}{fg=blue}
\colorlet{alert}{blue}
\colorlet{themecolor}{blue}

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

%  \definecolor{basecolor}{rgb}{0,1,.4}
%  \colorlet{themecolor}{basecolor!50!black}
% \definecolor{alertbasecolor}{rgb}{0,1,.4}
%  \colorlet{alert}{alertbasecolor!50!black}
%  \usecolortheme[named= themecolor]{structure}


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

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


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

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

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

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

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

\def\<{\langle} \def\>{\rangle}
\def\ad{\mathrm{ad}} 
\def\Aut{\mathrm{Aut}}
\def\Coind{\mathrm{Coind}} 
\def\dim{\mathrm{dim}} 
\def\End{\mathrm{End}} 
\def\ev{\mathrm{ev}} 
\def\half{\hbox{$\frac12$}}
\def\Hom{\mathrm{Hom}} 
\def\id{\mathrm{id}}
\def\img{\mathrm{img}}
\def\Ind{\mathrm{Ind}}
\def\ker{\mathrm{ker}}
\def\normeq{\unlhd}
\def\proj{\mathrm{proj}}
\def\qtr{\mathrm{qtr}} 
\def\str{\mathrm{str}} 
\def\tr{\mathrm{tr}} 
\def\Tr{\mathrm{Tr}} 
\def\vep{\varepsilon}
\def\Wedge{\hbox{$\bigwedge$}}
\def\Tk{\mathbf{K}}
\def\Tn{\mathcal{K}}

\def\brac{]}

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

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



%%%%%%%%%%%%%%%%
\usepackage{array}

\newsavebox\topalignbox
\newcolumntype{T}{%
  >{\begin{lrbox}\topalignbox
    \rule{0pt}{\ht\strutbox}}
  c
  <{\end{lrbox}%
    \raisebox{\dimexpr-\height+\ht\strutbox\relax}%
      {\usebox\topalignbox}}}
\newcolumntype{M}{>{$\vcenter\bgroup\hbox\bgroup}c<{\egroup\egroup$}}



%%%%%%%%%%%%%%%%%
\usepackage{tikz}
\usepgflibrary{shapes.geometric}
\tikzstyle{v}=[draw, fill =black, circle, inner sep=0pt, minimum size=1.5pt]
\tikzstyle{V}=[draw, fill =alert, circle, inner sep=0pt, minimum size=5pt]
\tikzstyle{C}=[draw, thick, fill =white, circle, inner sep=0pt, minimum size=5pt]
\tikzstyle{D}=[draw, black, fill =green!70!white, diamond, inner sep=0pt, minimum size=5pt]
\tikzstyle{T}=[draw, black, fill =alert, isosceles triangle, inner sep=0pt, minimum size=5pt] 
\tikzstyle{bT}=[draw, black, fill =alert, isosceles triangle, inner sep=0pt, minimum size=5pt, rotate = 180] 
\tikzstyle{bV}=[draw, fill =black, circle, inner sep=0pt, minimum size=4pt]
\tikzstyle{over}=[draw=white,double=black,line width=2pt, double distance=.75pt]
\tikzstyle{diagram}=[line width=.75pt, scale=\SCALE]

 \newcommand{\TikZ}[1]{\begin{matrix}\begin{tikzpicture}#1\end{tikzpicture}\end{matrix}}

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


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


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

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

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


\usepackage{graphicx}

\title{Representations of the \\
periplectic Lie superalgebra $\fp(n)$}
\author[Zajj Daugherty]{Zajj Daugherty\\
\small
Joint with M.\ Balagovic, I.\ Entova-Aizenbud,  I.\ Halacheva, \\
J.\ Hennig, M.\ S.\ Im, 
G.\ Letzter 
E.\ Norton,\\ 
V.\ Serganova, and  C.\ Stroppel \\~\\ arXiv:1610.08470}
\date{November 28, 2017}





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



%%%%%%%
\begin{frame}{Lie superalgebras}
A \alert{Lie superalgebra} is a  $\ZZ_2$-graded vector space $\mathfrak{g} =\mathfrak{g}_0 \oplus \mathfrak{g}_1$ with a \alert{super} Lie bracket \only<1-4>{
$$[,]:\fg \otimes \fg \to \fg$$
satisfying \\ \smallskip
\centerline{$\displaystyle
[x,y] = - (-1)^{\bar x\bar y}[y,x]
$} \smallskip
and\\ \smallskip
\centerline{$\displaystyle
[x,[y,z]]] = [[x,y], z] + (-1)^{\bar x\bar y}[y ,[x,z]],
$} \smallskip
where $x, y, z$ are each homogeneous, and $\bar{x}$ means \alert{degree}.
}\only<5->{
$[,]:\fg \otimes \fg \to \fg$ satisfying a super symmetry and super Jacobi identity.
}
\pause \medskip

Three types: \alert{basic}, \alert{Cartan type}, and \alert{strange} (two families: \alert{periplectic} and queer).

\medskip

\pause 
Let $V = V_0 \oplus V_1 = \CC^{m|n}$ be a $\ZZ_2$-graded vector space over $\CC$. \\
\only<5->{For (homogeneous) $v \in V_i$, write $\bar{v}=i$ for its \alert{degree}.}




\medskip\pause
The \alert{general linear Lie superalgebra is} \\ \smallskip
\centerline{$\displaystyle
\fgl(m|n) = \End(V)\pause = \fg_0 \oplus \fg_1,
$} \smallskip
where 
\\ \smallskip
\only<5->{\centerline{$\displaystyle
\fg_0 = \left\{\left. \begin{pmatrix} A & 0 \\ 0 & D\end{pmatrix} ~\right|~ A \in \End(V_0), D \in \End(V_1)\right\},
$} 
\smallskip
\centerline{$\displaystyle
\fg_1 = \left\{\left. \begin{pmatrix} 0 & B \\ C & 0\end{pmatrix} ~\right|~ B \in \Hom(V_1,V_0), C \in  \Hom(V_0,V_1)\right\}.
$}} \smallskip


\pause 
\alert{Bracket:} $[x,y] = xy - (-1)^{\bar x\bar y}yx$.

\end{frame}

%%%%%%%
\begin{frame}{Lie superalgebras}

Let $V = V_0 \oplus V_1 = \CC^{m|n}$ be a $\ZZ_2$-graded vector space over $\CC$. \\
For homogeneous $v$, write $\bar{v}$ for its \alert{degree}.\\
$\fgl(m|n) = \End(V)$ with bracket $[x,y] = xy - (-1)^{\bar x\bar y}yx$.

\medskip\pause 

Let $\beta: V \otimes V \to \CC$ be a nondegenerate, homogeneous, bilinear form satisfying \\ \smallskip
\centerline{$\displaystyle
\beta(v,w) = (-1)^{\bar{v}\bar{w}}\beta(w,v)\quad  \text{(\alert{supersymmetric})}.
$} \smallskip
\pause 

Then 
$$\fg = \{x \in \End(V) ~|~ \beta(xv,w) + (-1)^{\bar{x}\bar{v}}\beta(v,xw)=0\}$$
is a Lie superalgebra  ($\ZZ_2$-graded). \pause  

\medskip

\alert{Ex.} If $\beta$ is \alert{even}, $\fg=\mathfrak{osp}(V)$ the \alert{orthosymplectic} Lie superalgebra  (if $V_1 = 0$, $\fg = \fso(V)$; and if $V_0 = 0$, $\fg = \fsp(V) $).\pause

\medskip

\alert{Ex.} If $\beta$ is \alert{odd}, we need $m=n$. Then $\fg=\fp(V)= \fp(n)$ the \alert{periplectic} Lie superalgebra.


\end{frame}



%%%%%%%
\begin{frame}{Periplectic Lie superalgebra}
Let $V = V_0 \oplus V_1 = \CC^{n|n}$ be a $\ZZ_2$-graded vector space over $\CC$. \\
If $\beta: V \otimes V \to \CC$ is an odd, nondegenerate, homogeneous, super symmetric bilinear form, then 
\\ \smallskip
\centerline{$\beta(v,w) = 0 \quad \text{ if } \quad \bar v = \bar  w.
$}

\pause \medskip


The \alert{periplectic} Lie superalgebra is \\ \smallskip
\centerline{$\displaystyle
\fp(V) = \fp(n)= \{x \in \End(V) ~|~ \beta(xv,w) + (-1)^{\bar{x}\bar{v}}\beta(v,xw)=0\}.
$}

\pause \medskip

Specifically, we have 
\\ \smallskip
\centerline{$\displaystyle
\fp(n) \cong \left\{\left. \begin{pmatrix} A & B \\ C & -A^t \end{pmatrix} \in \fgl(n|n) 
		~\right|~ B = B^t, C = -C^t \right\}.
$} \smallskip \pause 
Then, as vector spaces $\fp(n) = \fg_0 \oplus \fg_1 \oplus \fg_{-1} $, where  
\\ \smallskip
\centerline{$\displaystyle
\fg_0 = \left\{ \begin{pmatrix} A & 0 \\ 0 & -A^t \end{pmatrix}\right\} \cong \fgl_n
$} \smallskip
\centerline{$\displaystyle
\fg_1  = \left\{ \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix}\right\}, \quad 
 \fg_{-1} = \left\{ \begin{pmatrix} 0 & 0 \\ C & 0 \end{pmatrix}\right\}.
$} \smallskip

\pause
\alert{Goal:} Study the representation theory of $\fp(n)$. \pause In particular,  study the category $\mathcal{F}_n$ of finite-dimensional integrable representations.
\end{frame}




\begin{frame}
Let $\mathcal{F}_n$ be the category of finite-dimensional integrable representations.

\alert{Fact:} There is no duality functor that preserves simple objects. \pause Although projective modules are also injective and tilting, the  standard and costandard modules have different combinatorics! Call these 
\alert{thick} and \alert{thin Kac modules}, respectively.

\pause \medskip

\alert{Weights:} The cartan is the set of diagonal matrices of $\fg_0 \cong \fgl(n)$. So the dominant weights are indexed by \\ \smallskip
\centerline{$\displaystyle
\lambda = \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n, \quad \lambda_i \in \ZZ.
$} \smallskip \pause 


\medskip

\alert{Highest weight modules:} Let $V(\lambda)$ and  $L(\lambda)$ be the simple $\fg_0$ and $\fp(n)$-modules, respectively, of highest weight $\lambda$.
\pause \medskip


Define the \alert{thick Kac module} (corresponding to $\lambda$) as
\\ \smallskip
\centerline{$\displaystyle
\Tk(\lambda)=\operatorname{Ind}^{\fp(n)}_{\fg_0\oplus\fg_{-1}}V(\lambda), 
$} \smallskip
and the \alert{thin Kac module} (corresponding to $\lambda$) as\\ \smallskip
\centerline{$\displaystyle
\Tn(\lambda)=\Coind^{\fp(n)}_{\fg_0\oplus\fg_1}V(\lambda).
$} \smallskip
 \hfill {\footnotesize ($\dim(\Tk(\lambda)) = 2^n \dim(\Tn(\lambda))$)}


\end{frame}

\begin{frame}
The category $\cF_n$ is a \alert{highest weight category} {\scriptsize(in the sense of Cline, Parshall, and Scott)}, with two natural highest weight structures:   
\begin{enumerate}[1.]
\item With standard modules $\Tk(\lambda)$, costandard modules $\Tn(\lambda)$, and  partial order on dominant weights given by
\\ \smallskip
\centerline{
 $\mu \geq \lambda$  \quad whenever  \quad $\mu_i \leq \lambda_i$ for each $i$;
} \smallskip
\item With standard modules $\Tn(\lambda)$ and costandard modules $\Tk(\lambda)$.
\end{enumerate}


\pause 

\begin{thm} \vspace{-.1in}
\begin{enumerate}[(1)]
	\item (\alert{BGG reciprocity}) The multiplicity of $\Tk(\lambda)$ in $P(\lambda)$ (as summands) is equal to the  multiplicity of $L(\lambda)$ in $\Tn(\lambda)$ (as subquotients), and vice versa (with a shift). 
	\item  If  $\Tk(\mu)$ appears in $P(\lambda)$, then $\mu \geq \lambda$.
	\item We have: $\Tk(\lambda)\cong P(\lambda)$ and $\Tn(\lambda)\cong L(\lambda)$ if and only if $\lambda$ has distinct parts. [Kac78]
\end{enumerate}
\end{thm}

\vspace{-.1in}
\pause 
{[\alert{Disclaimer:} there is a parity-switching involution $\Pi$ on $\fp(V)$ and all of its modules. Many of these and the following results are technically ``$M$ or $\Pi^\ell M$'' or ``$\Pi^\ell \phi$''  in place of an operator $\phi$. See paper for details.] }

\end{frame}


\begin{frame}

\alert{Key ingredients for other cases:} a large center in  $\mathcal{U}(\mathfrak g)$, and  translation functors given by tensoring with the natural representation (respectively its dual) followed by the projection onto a block.\pause \medskip


\alert{Issue here:} The center of $\mathcal{U}(\mathfrak {p}(n))$ is  trivial! \pause \medskip

\alert{Key observation:} With respect to the inner product\\ \smallskip
\centerline{$\displaystyle
\<x,y\> = \str(xy) = \tr\left(xy|_{V_0}\right) - \tr\left(xy|_{V_1}\right),
$} \smallskip
on $\fgl(n|n)$, we have 
\\ \smallskip
\centerline{$\displaystyle
\fp(n)^* = \fp(n)^\perp \qquad \text{so} \qquad \fgl(n|n) = \fp(n) \oplus \fp(n)^*.
$} \smallskip
\pause We define the \alert{split (fake) Casimir} element of \\ \smallskip
\centerline{$\displaystyle
\fp(n) \otimes \fp(n)^* \subset \fp(n) \otimes\mathfrak{gl}(n|n)
$} \smallskip by 
\\ \smallskip
\centerline{$\displaystyle
\Omega=2\sum b\otimes b^* ,
$} \smallskip
where the sum is over a basis of $\fp(n)$ and its dual in $\fp(n)^*$. \pause \medskip

Then for any $\fp(n)$-module $M$, $\Omega$ acts on $M \otimes V$. \pause\\ Moreover, it commutes with the action of $\fp(n)$. 
\end{frame}


\begin{frame}{Example: $V \otimes V$}
As a $\fp(V)$-module, \\ \smallskip
\centerline{$\displaystyle
V\otimes V = S^2 V\oplus\Lambda^2V
$} \smallskip\pause 
Let $\{v_i ~|~ i = 1, \dots, n\}$ and $\{v_{i'} ~|~ i = 1, \dots, n\}$ be orthogonal bases of $V_0$ and $V_1$, with $\beta(v_i, v_{j'}) = \delta_{ij}$. Then 
\\ \smallskip
\centerline{$\displaystyle
c= \sum_{i}(v_i\otimes v_{i'}-v_{i'}\otimes v_{i})\in\Lambda^2 V
$} \smallskip
spans the unique trivial 
submodule in $V\otimes V$. \pause 

\medskip

The algebra $\End_{\fg}(V\otimes V)$ is 3-dimensional with basis $1$, 
\\ \smallskip
\centerline{$\displaystyle
s: v\otimes w\mapsto (-1)^{p(v)p(w)} w\otimes v, \quad \text{and} \quad 
e={\beta^*}\circ\beta: v\otimes w \mapsto \beta(v,w)c.
$} \smallskip

\medskip\pause

Draw: {\def\SCALE{.65}
\\ \smallskip
\centerline{
$\phantom{\alert{marked Brauer}}$\hfill
$\displaystyle
s  = \TikZ{[diagram]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\draw (t1) to (b2) (t2) to (b1);
}
\quad \text{ and } \quad 
e = \TikZ{[diagram, xscale=1.5]
\foreach\x in {1, 2}{\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\only<1-6>{\draw (t1) to [bend right=50] (t2) (b2) to [bend right=50] (b1);}
\only<6->{\draw (t1) to [bend right=50] node[bT]{} (t2) (b2) to [bend right=50] node[D]{} (b1);}
}\ .
$ \qquad \uncover<6->{(\alert{marked Brauer})}}} 

\medskip\pause

Note that  $e=e\circ s=-s\circ e$. \pause \pause Also, $e^2 = 0$. \hfill(non-semisimple case)

\medskip\pause

Then, on $V \otimes V$,  we have $\Omega = s + e$. \hfill {\scriptsize(same as classical case)}
\end{frame}


\begin{frame}{Translation functors}
Consider the following endofunctor of $\cF_n$,
\\ \smallskip
\centerline{$\displaystyle
\Theta'= \underline{~~}\otimes V:\quad\cF_n\longrightarrow\cF_n.
$} \smallskip 
(Exact, with left- and right-adjoint $\Pi \Theta$.)\pause

\medskip


Again\\ \smallskip
 \centerline{$\displaystyle
\Omega=2\sum b\otimes b^* \quad $ acts on $ M \otimes V$
}\smallskip
for any $\fp(V)$-module $M$, and that action commutes with the action of $\fp(V)$.  \pause So $\Omega$ gives a natural endotransformation of $\Theta'$. \pause \medskip



\only<1-5>{
Similarly, let $\Omega_{i, j}$ be the operator $\Omega$ acting on the $i$ and $j$ tensor factor of $M \otimes V^d$ (were $M$ is the 0th factor). Then $\Omega_{i,j}$ is an endomorphism of the endofunctor $ \underline{~~}\otimes V^{\otimes d}$.

\medskip}\pause 

\only<1-5>{If \\ \smallskip
\centerline{$\displaystyle
y_j = \sum_{i<j} \Omega_{i,j} = (\Delta^{j}\otimes 1)(\Omega) \otimes 1^{d-j},
$} \smallskip
then $y_1, y_2, \dots, y_d$ form a commutative family of endomorphisms of 
\\ \smallskip
\centerline{$\displaystyle
 \underline{~~}\otimes V^{\otimes d}:\quad\cF_n\longrightarrow\cF_n.
$} 

\medskip}\pause 

Let \alert{$\Theta_k '= \proj_k \Theta$}, where $\proj_k$ is the projection onto the $k$ eigenspace of $\Omega$. 

 \pause 

\begin{thm}
 $\Theta_k'$ is exact, and is 0 for $k \notin \ZZ$. So $\Theta' = \bigoplus_{k \in \ZZ} \Theta_k'$.
 \end{thm}
 

\end{frame}







\begin{frame}{Weight diagrams}
For a dominant weight $\lambda$, let $\bar\lambda = \lambda + \rho$, \\
\hfill where $\rho = (n-1, n-2, \dots, 0)$. \pause 

Let $c_{\lambda}:=\{\bar{\lambda_i}\,|\,i=1, \ldots, n\}$. \pause \smallskip

\alert{Weight diagrams.} Draw a sequence of filled and empty circles, one for each integer, where \\ 
\centerline{
$\TikZ{\node[V, label = below:{\tiny $i$}, label = above:{\tiny $\phantom{i}$}] at (0,0) {};}
$ means $i \in c_\lambda$ \quad 
$\TikZ{\node[C, label = below:{\tiny $i$}, label = above:{\tiny $\phantom{i}$}] at (0,0) {};}
$ means $i \notin c_\lambda$}  \pause

\alert{Ex. } $n=4, \lambda = 0$:\\
\centerline{$\TikZ{\foreach \x in {-1, 4, 5, 6, 7}{\node[C, label = below:{\tiny $\x$}] at (\x,0) {};}
		\foreach \x in {0, 1, 2, 3}{\node[V, label = below:{\tiny $\x$}] at (\x,0) {};}
		\node at (-1.5,0) {$\cdots$}; \node at (7.5,0) {$\cdots$}; }$} \pause

 \alert{Ex. } $n=4, \lambda = \rho = (3,2,1,0)$:\\
\centerline{$\TikZ{\foreach \x in {-1, 1, 3, 5, 7}{\node[C, label = below:{\tiny $\x$}] at (\x,0) {};}
		\foreach \x in {0, 2, 4, 6}{\node[V, label = below:{\tiny $\x$}] at (\x,0) {};}
		\node at (-1.5,0) {$\cdots$}; \node at (7.5,0) {$\cdots$}; }$}
\medskip\pause 

\alert{Fact:} $\lambda$ has distinct parts iff every pair of filled dots has at least one empty dot separating them.  \hfill {\footnotesize($\bar{\lambda}$ \emph{always} has dist.\ parts)}\pause 


\alert{Fact:} $\lambda \leq \mu$ iff the $i$-th filled circle in $\lambda$ is to the right of the $i$-th filled circle in $\mu$.


\end{frame}



\begin{frame}{Translations on thick Kac modules}
\alert{\Large Theorem.} $\Theta'_k$ on $\Tk(\lambda)$ is given by\dots \pause 
\begin{enumerate}[(1)]
\item $\Theta'_k(\Tk(\lambda)) = \Tk(\mu)$ whenever the $k-2$, $k-1$, and $k$ positions of $\lambda$ and $\mu$ look like \\\smallskip
\centerline{$\TikZ{
		\node[left] at (.5,0){$\lambda:$};
		\node[V, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[C, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[C, label = below:{\tiny $k$}] at (3,0) {};} \qquad\qquad 
		%
		\TikZ{
		\node[left] at (.5,0){$\mu:$};
		\node[C, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[V, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[C, label = below:{\tiny $k$}] at (3,0) {};}
		$}\pause 

\item $\Theta'_k(\Tk(\lambda)) = \Pi\Tk(\mu)$ whenever the $k-2$, $k-1$, and $k$ positions of $\lambda$ and $\mu$ look like \\\smallskip
\centerline{$\TikZ{
		\node[left] at (.5,0){$\lambda:$};
		\node[C, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[C, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[V, label = below:{\tiny $k$}] at (3,0) {};} \qquad\qquad 
		%
		\TikZ{
		\node[left] at (.5,0){$\mu:$};
		\node[C, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[V, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[C, label = below:{\tiny $k$}] at (3,0) {};}
		$}\pause

\item There is a short exact sequence \\ \smallskip
\centerline{$\displaystyle
0 \to \Pi\Tk(\mu) \to \alert{\Theta'_k \Tk(\lambda)} \to \Tk(\mu') \to 0
$} 
whenever the $k-2$, $k-1$, and $k$ positions of $\mu$, $\lambda$, and  $\mu'$ look like \\\smallskip
\centerline{$\TikZ{
		\node[left] at (.75,0){$\mu:$};
		\node[V, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[V, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[C, label = below:{\tiny $k$}] at (3,0) {};}
		$  \qquad$\TikZ{
		\node[left] at (.75,0){$\lambda:$};
		\node[V, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[C, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[V, label = below:{\tiny $k$}] at (3,0) {};}$ 
		 \qquad  $\TikZ{
		\node[left] at (.75,0){$\mu':$};
		\node[C, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[V, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[V, label = below:{\tiny $k$}] at (3,0) {};}
		$}\pause
\end{enumerate}
Otherwise, $\Theta'_k(\Tk(\lambda)) = 0$.
\end{frame}





\begin{frame}{Translations on thin Kac modules}
\alert{\Large Theorem.} $\Theta'_k$ on $\Tn(\lambda)$ is given by\dots \pause
\begin{enumerate}[(1)]
\item $\Theta'_k(\Tn(\lambda)) = \Tn(\mu)$ whenever the $k-1$, $k$, and $k+1$ positions of $\lambda$ and $\mu$ look like \\\smallskip
\centerline{$\TikZ{
		\node[left] at (.5,0){$\lambda:$};
		\node[V, label = below:{\tiny $k-1$}] at (1,0) {};
		\node[V, label = below:{\tiny $k$}] at (2,0) {};
		\node[C, label = below:{\tiny $k+1$}] at (3,0) {};} \qquad\qquad 
		%
		\TikZ{
		\node[left] at (.5,0){$\mu:$};
		\node[V, label = below:{\tiny $k-1$}] at (1,0) {};
		\node[C, label = below:{\tiny $k$}] at (2,0) {};
		\node[V, label = below:{\tiny $k+1$}] at (3,0) {};}
		$}\pause 

\item $\Theta'_k(\Tn(\lambda)) = \Pi\Tn(\mu)$ whenever the $k-1$, $k$, and $k+1$ positions of $\lambda$ and $\mu$ look like \\\smallskip
\centerline{$\TikZ{
		\node[left] at (.5,0){$\lambda:$};
		\node[C, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[V, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[V, label = below:{\tiny $k$}] at (3,0) {};} \qquad\qquad 
		%
		\TikZ{
		\node[left] at (.5,0){$\mu:$};
		\node[V, label = below:{\tiny $k-1$}] at (1,0) {};
		\node[C, label = below:{\tiny $k$}] at (2,0) {};
		\node[V, label = below:{\tiny $k+1$}] at (3,0) {};}
		$}\pause

\item There is a short exact sequence \\ \smallskip
\centerline{$\displaystyle
0 \to \Pi\Tn(\mu) \to \alert{\Theta'_k \Tn(\lambda)} \to \Tn(\mu') \to 0
$} 
whenever the $k-1$, $k$, and $k+1$ positions of $\mu$, $\lambda$, and  $\mu'$ look like \\\smallskip
\centerline{$\TikZ{
		\node[left] at (.75,0){$\mu:$};
		\node[V, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[C, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[C, label = below:{\tiny $k$}] at (3,0) {};}
		$  \qquad $\TikZ{
		\node[left] at (.75,0){$\lambda:$};
		\node[C, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[V, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[C, label = below:{\tiny $k$}] at (3,0) {};}$ 
		 \qquad  $\TikZ{
		\node[left] at (.75,0){$\mu':$};
		\node[C, label = below:{\tiny $k-2$}] at (1,0) {};
		\node[C, label = below:{\tiny $k-1$}] at (2,0) {};
		\node[V, label = below:{\tiny $k$}] at (3,0) {};}
		$}\pause
\end{enumerate}
Otherwise, $\Theta'_k(\Tn(\lambda)) = 0$.
\end{frame}




\begin{frame}{Algebraic structure of translation functors}
Define $\Theta_k = \Pi^k \Theta_k'$ and $\Theta = \bigoplus_{k \in \ZZ} \Theta_k$. \pause 

\begin{thm}
 The functor $\Theta_k$ is exact, and is \\ \smallskip
\centerline{left adjoint to $ \Theta_{k-1}$ \quad and \quad right adjoint to $\Theta_{k+1}$.}
 \end{thm}


\pause 


\begin{thm} Up to natural isomorphisms (e.g.\ on the Grothendeik group), we have 
\\ \smallskip
\centerline{$\displaystyle
   \Theta_k^2= 0, \qquad 
  \Theta_k \Theta_j = \Theta_j \Theta_k,
$ \quad for $|j-k|>1$,}
  \smallskip
  \centerline{ and \qquad $\displaystyle
   \Theta_k \Theta_{k \pm 1} \Theta_k = \Theta_k.
$} \smallskip
Namely, the functors $\Theta_k$ generate the infinite Temperley-Lieb algebra $TL_\infty(0)$ (the parameter is  $q+q^{-1}$, where $q = i$).  

\end{thm}

\pause 

(The maps on the Grothendieck group induced by the $\Theta'_k$'s also satisfy these relations. At the level of $\cF_n$, though, it's the $\Theta_k$ that we want.)

\end{frame}


\begin{frame}

\alert{\Large Other results:}
\begin{enumerate}[{$\blacktriangleright$}]
\item Computations of decompositions/filtrations of projectives in terms of Kac modules, and Kac modules in terms of simples. \pause 
\item  Translation functors applied to projectives and simples.\pause
\item Multiplicity free results. \pause
\item Socles and cosocles of Kac modules. \pause
\item Blocks of $\cF_n$. 
\end{enumerate}
\pause 

\bigskip

\alert{\Large Next paper:} 
\begin{enumerate}[{$\blacktriangleright$}]
\item Cyclotomic marked Brauer algebras.
\end{enumerate}


\end{frame}





%%%%%%%%%%%
\begin{frame}



\def\SCALE{.75}


(Moon 2003, Kujawa-Tharp 2014) The \alert{marked Brauer algebra} $B_d(\delta, \epsilon)$, $\epsilon = \pm 1$, is the space spanned by \alert{marked Brauer diagrams}
$$d=\TikZ{[diagram]
\foreach\x in {1,2,...,5}{
\node[v] (t\x) at (\x,1.5){};\node[v] (b\x) at (\x,0){};}
\node[v] (t6) at (6,1.5){};\node[v] (b6) at (6,0){};
\draw (t1) to (b2) (t4) to (b6)
	(t2) to [bend right] node[bT]{} (t6) (t3) to [bend right] node[T]{} (t5)
	(b4) to [bend right=40] node[D]{} (b1) (b3) to [bend left] node[D]{} (b5);
}
\qquad 
\alert{\begin{array}{l}
\text{caps get one $\TikZ{[diagram]\node[D] at (0,0){};}$ each,}\\
\text{cups get one $\TikZ{[diagram]\node[T] at (0,0){};}$ or $\TikZ{[diagram]\node[bT] at (0,0){};}$ each, }\\
\text{no two markings at same height.}
\end{array}}
$$
 with equivalence up to isotopy except for the local relations 

\smallskip

\centerline{$\begin{matrix}\TikZ{[diagram]\draw (0,0) to [bend right=40] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}\\
\TikZ{[diagram]\draw[rounded corners] (0,0) to (.75,-.25) to [bend left=40] (0,-.25) to (.75,0); \node[T] at (.75/2, -.4) {}; \draw[thin, densely dotted] (0,-.6) rectangle (.75,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT] {}(1,0); \draw[thin, densely dotted] (0,-.4) rectangle (1,.1);}
\end{matrix} 
\quad \text{and}\quad
\TikZ{[diagram]
\draw (0,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,.8); \draw[thin, densely dotted] (0,.5) rectangle (1,1.1);
\draw (1.25,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,0);  \draw[thin, densely dotted] (1.25,.3) rectangle (2.25,-.3);}
=
\epsilon\ 
\TikZ{[diagram]
\draw (0,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,0); \draw[thin, densely dotted] (0,.3) rectangle (1,-.3);
\draw (1.25,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,.8);  \draw[thin, densely dotted] (1.25,.5) rectangle (2.25,1.1);}
$}

\smallskip

for any \alert{adjacent} markings $\TikZ{[diagram] \node[draw, fill=white, circle, inner sep=1pt] at (0,0) {\scriptsize$x$};}$ and $\TikZ{[diagram] \node[draw, fill=white, circle, inner sep=1pt] at (0,0) {\scriptsize$y$};}$ (meaning no markings of height between these two). \pause Multiplication is given by vertical concatenation, with relations  
$\TikZ{[line width=.75pt, scale=.4] \draw (1,1) ellipse [x radius=.75cm,y radius=.5cm];}=\delta,$

$$\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[T]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}
\quad \text{ and } \quad 
\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \epsilon\ \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[bT]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}\ .$$



\end{frame}



%%%%%%
\begin{frame}

\def\SCALE{.75}

$$\begin{matrix}\TikZ{[diagram]\draw (0,0) to [bend right=40] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}\\
\TikZ{[diagram]\draw[rounded corners] (0,0) to (.75,-.25) to [bend left=40] (0,-.25) to (.75,0); \node[T] at (.75/2, -.4) {}; \draw[thin, densely dotted] (0,-.6) rectangle (.75,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT] {}(1,0); \draw[thin, densely dotted] (0,-.4) rectangle (1,.1);}
\end{matrix}  
\qquad
\TikZ{[diagram]
\draw (0,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,.8); \draw[thin, densely dotted] (0,.5) rectangle (1,1.1);
\draw (1.25,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,0);  \draw[thin, densely dotted] (1.25,.3) rectangle (2.25,-.3);}
=
\epsilon\ 
\TikZ{[diagram]
\draw (0,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,0); \draw[thin, densely dotted] (0,.3) rectangle (1,-.3);
\draw (1.25,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,.8);  \draw[thin, densely dotted] (1.25,.5) rectangle (2.25,1.1);}
$$
$$\TikZ{[line width=.75pt, scale=.4] \draw (1,1) ellipse [x radius=.75cm,y radius=.5cm];}=\delta$$
$$\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[T]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}
\qquad
\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \epsilon\ \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[bT]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}$$






\pause\bigskip

For example, \def\SCALE{.6}
$$\TikZ{[diagram]
\foreach\x in {1,2,...,6}{
\node[v] (t\x) at (\x,1.6){};\node[v] (b\x) at (\x,.1){};}
\draw (t1) to (b2) (t4) to (b6)
	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5)
	(b4) to [bend right=40] node[D]{} (b1) (b3) to [bend left] node[D, pos=.6, sloped]{} (b5);
\foreach\x in {1,2,...,6}{
\node[v] (bt\x) at (\x,-.1){};\node[v] (bb\x) at (\x,-1){};
\draw[thin] (bt\x) to (b\x);}
\draw (bt1) to (bb1) (bt2) to (bb2) (bt5) to (bb5) (bt6) to (bb6)
	(bt3) to [bend right] node[T]{} (bt4) (bb4) to [bend right] node[D]{} (bb3)
	;}\pause
= \epsilon\ 
\TikZ{[diagram]
\foreach\x in {1,2,...,6}{
\node[v] (t\x) at (\x,1.5){};\coordinate (b\x) at (\x,0); \node[v] (bb\x) at (\x,-.9){};}
\draw 
	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5) (bb4) to [bend right] node[D]{} (bb3);
\draw [rounded corners] (t1) to (b2) to (bb2)
	(t4) to (b6) to (bb6)
	 (bb1) to (b1) to [bend left=50] node[pos=.6] (a){} (b3)  to [bend right] node (b){} (b4) to [bend left] node (c){} (b5) to (bb5);
\draw[densely dotted, red] (3,-.5) rectangle (5,.5);
\node[D] at (a){}; \node[bT] at (b){}; \node[D] at (c){};
}\pause
=
\TikZ{[diagram]
\foreach\x in {1,2,...,6}{
\node[v] (t\x) at (\x,1.5){};\node[v]  (b\x) at (\x,0){};}
\draw 
	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5) 
	(b4) to [bend right] node[D]{} (b3) (b5) to [bend right] node[D]{} (b1)
	(t1) to (b2) (t4) to [bend left] (b6);
}
$$
%\pause Alternatively,
%$$
%= \epsilon\ 
%\TikZ{[diagram]
%\foreach\x in {1,2,...,6}{
%\node[v] (t\x) at (\x,1.5){};\coordinate (b\x) at (\x,0); \node[v] (bb\x) at (\x,-.9){};}
%\draw 
%	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5) (bb4) to [bend right] node[D]{} (bb3);
%\draw [rounded corners] (t1) to (b2) to (bb2)
%	(t4) to (b6) to (bb6)
%	 (bb1) to (b1) to [bend left=50] node[pos=.6] (a){} (b3)  to [bend right=40] node (b){} (b4) to [bend left=40] node (c){} (b5) to (bb5);
%\draw[densely dotted, red] (2,0) rectangle (4.75,.7);
%\node[D] at (a){}; \node[bT] at (b){}; \node[D] at (c){};
%}\pause =
%\TikZ{[diagram]
%\foreach\x in {1,2,...,6}{
%\node[v] (t\x) at (\x,1.5){};\coordinate (b\x) at (\x,0); \node[v] (bb\x) at (\x,-.9){};}
%\draw 
%	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5) (bb4) to [bend right] node[D]{} (bb3);
%\draw [rounded corners] (t1) to (b2) to (bb2)
%	(t4) to (b6) to (bb6)
%	 (bb1) to (b2) to [bend left=20] node[pos=.6] (a){} (b3)  to [bend right=50] node (b){} (b4) to [bend left=50] node (c){} (b5) to (bb5);
%%\draw[densely dotted, red] (2,0) rectangle (4.75,.7);
%\node[D] at (a){}; \node[bT] at (b){}; \node[D] at (c){};
%}\pause
%=
%\TikZ{[diagram]
%\foreach\x in {1,2,...,6}{
%\node[v] (t\x) at (\x,1.5){};\node[v]  (b\x) at (\x,0){};}
%\draw 
%	(t2) to [bend right] node[bT, pos=.4]{} (t6) (t3) to [bend right] node[T]{} (t5) 
%	(b4) to [bend right] node[D]{} (b3) (b5) to [bend right] node[D]{} (b1)
%	(t1) to (b2) (t4) to [bend left] (b6);
%%	 (bb1) to (b1) to [bend left=40] (b4)  to [bend left] (b3) to [bend left] (b5) to (bb5);
%%\node[T] at (2.5,.7){};
%} \checkmark
%$$
\end{frame}






%%%%%%%%
\begin{frame}

\def\SCALE{.75}


(Moon 2003, Kujawa-Tharp 2014) The \alert{marked Brauer algebra} $B_d(\delta, \epsilon)$, $\epsilon = \pm 1$, is the space spanned by \alert{marked Brauer diagrams}
$$d=\TikZ{[diagram]
\foreach\x in {1,2,...,5}{
\node[v] (t\x) at (\x,1.5){};\node[v] (b\x) at (\x,0){};}
\node[v] (t6) at (6,1.5){};\node[v] (b6) at (6,0){};
\draw (t1) to (b2) (t4) to (b6)
	(t2) to [bend right] node[bT]{} (t6) (t3) to [bend right] node[T]{} (t5)
	(b4) to [bend right=40] node[D]{} (b1) (b3) to [bend left] node[D]{} (b5);
}
\qquad 
\alert{\begin{array}{l}
\text{caps get one $\TikZ{[diagram]\node[D] at (0,0){};}$ each,}\\
\text{cups get one $\TikZ{[diagram]\node[T] at (0,0){};}$ or $\TikZ{[diagram]\node[bT] at (0,0){};}$ each, }\\
\text{no two markings at same height.}
\end{array}}
$$
with equivalence up to isotopy except for the local relations 

\smallskip

\centerline{$\begin{matrix}\TikZ{[diagram]\draw (0,0) to [bend right=40] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.5) rectangle (1.5,.1);}\\
\TikZ{[diagram]\draw[rounded corners] (0,0) to (.75,-.25) to [bend left=40] (0,-.25) to (.75,0); \node[T] at (.75/2, -.4) {}; \draw[thin, densely dotted] (0,-.6) rectangle (.75,.1);}
=
\epsilon\ \TikZ{[diagram]\draw (0,0) to [bend right=40] node[bT] {}(1,0); \draw[thin, densely dotted] (0,-.4) rectangle (1,.1);}
\end{matrix} 
\quad \text{and}\quad
\TikZ{[diagram]
\draw (0,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,.8); \draw[thin, densely dotted] (0,.5) rectangle (1,1.1);
\draw (1.25,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,0);  \draw[thin, densely dotted] (1.25,.3) rectangle (2.25,-.3);}
=
\epsilon\ 
\TikZ{[diagram]
\draw (0,0) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$x$} (1,0); \draw[thin, densely dotted] (0,.3) rectangle (1,-.3);
\draw (1.25,.8) to node[draw, fill=white, circle, inner sep=1pt]{\scriptsize$y$} (2.25,.8);  \draw[thin, densely dotted] (1.25,.5) rectangle (2.25,1.1);}
$}

\smallskip

for any \alert{adjacent} markings $\TikZ{[diagram] \node[draw, fill=white, circle, inner sep=1pt] at (0,0) {\scriptsize$x$};}$ and $\TikZ{[diagram] \node[draw, fill=white, circle, inner sep=1pt] at (0,0) {\scriptsize$y$};}$ (meaning no markings of height between these two). Multiplication is given by vertical concatenation, with relations  
$\TikZ{[line width=.75pt, scale=.4] \draw (1,1) ellipse [x radius=.75cm,y radius=.5cm];}=\delta,$

\smallskip

\centerline{$
\begin{matrix}
\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[bT]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[T]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}\\
\text{and}\\
\TikZ{[diagram]\draw (0,0) to [bend left] node[D]{} (.75,0) to [bend right] node[T]{} (1.5,0); \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);} 
= \epsilon\ \TikZ{[diagram]\draw (0,0) to (.75,0); \draw[thin, densely dotted] (0,-.3) rectangle (.75,.3);}
=\TikZ{[diagram]\draw (0,0) to [bend right] node[bT]{} (.75,0) to [bend left] node[D]{} (1.5,0);  \draw[thin, densely dotted] (0,-.3) rectangle (1.5,.3);}\ .
\end{matrix}$
\quad
\fbox{\begin{minipage}{5.5cm}
\alert{Note:}\\
\alert{(1)} $B_d(\delta, 1) = B_d(\delta)$.\\
\alert{(2)} If $\epsilon = -1$, then multiplication is well-defined exactly when $\delta =0$.
\end{minipage}}
}



\end{frame}




\begin{frame}{Centralizer algebras}
\def\SCALE{.5}

The marked Brauer algebra $B_d(\delta, \epsilon)$ is generated by 
{\def\SCALE{.65}
\centerline{$s_i = \TikZ{[diagram]
\foreach\x in {1, 3, 4, 6}{
\node[v] (t\x) at (.75*\x,1){};\node[v] (b\x) at (.75*\x,0){};}
\draw (t1) to (b1) (t3) to (b4) (t4) to (b3) (t6) to (b6);
\node[above] at (t3) {\tiny\alert{$\phantom{+}i\ \phantom{+}$}};\node[above] at (t4) {\tiny\alert{$\ i\!+\!1$}};\node[below, white] at (b4) {\tiny{$\ i\!+\!1$}};
\node at (2*.75, .5){$\dots$}; \node at (5*.75, .5){$\dots$}; 
}
\quad \text{ and } \quad 
e_i = \TikZ{[diagram]
\foreach\x/\y in {1.5/1, 3/3, 4/4, 5.5/6}{\node[v] (t\y) at (\x,1){};\node[v] (b\y) at (\x,0){};}
\draw (t1) to (b1) (t3) to [bend right=40] node[bT]{} (t4) (b4) to [bend right] node[D]{} (b3) (t6) to (b6);
\node[above] at (t3) {\tiny\alert{$\phantom{+}i\ \phantom{+}$}};\node[above] at (t4) {\tiny\alert{$\ i\!+\!1$}};\node[below, white] at (b4) {\tiny{$\ i\!+\!1$}};
\node at (2+.25, .5){$\dots$}; \node at (5-.25, .5){$\dots$}; 
}\ ,
$}}
for $i=1, \dots, k-1$. 

\pause\medskip



Back in our more general setting $\fg = \fgl(V)^\beta$ with $V = V_0 \oplus V_1$ (for $\beta$ non-deg, homog, super symmetric, bilinear). \pause If  \\ \smallskip
\centerline{$\displaystyle
\beta^* : \CC \to V \otimes V
 \quad \text{ and } \quad 
	\begin{array}{rl} s: V \otimes V &\to V \otimes V \\
				u \otimes v &\mapsto (-1)^{\bar{u}\bar{v}} v \otimes u\pause,\end{array}
$} \smallskip
then the map	\\ \smallskip
\centerline{$\displaystyle
e_i \mapsto 1^{\otimes i-1} \otimes \beta^* \beta \otimes 1^{k-i-1}, \quad 
	s_i \mapsto 1^{\otimes i-1} \otimes s \otimes 1^{k-i-1},
$} \smallskip
for $ i=1, \dots, k-1$,
 gives 
\\ \smallskip
\centerline{$\displaystyle
B_d(\delta, \epsilon) \twoheadrightarrow \End_{\fg}(V^{\otimes d})
$} \smallskip
 when $\delta = \dim V_0 - \dim V_1$ and $\epsilon = (-1)^{\bar{\beta}}$ [KT14].




\end{frame}







\begin{frame}{Jucys-Murphy elements and the Casimir}
For $i<j$, let 
{\def\SCALE{.65}
\centerline{$s_{i,j} = \TikZ{[diagram]
\foreach\x/\y in {1/1, 2/2, 2.5/3, 4.5/4, 5/5, 6/6}{\node[v] (t\y) at (\x,1){};\node[v] (b\y) at (\x,0){};}
\draw (t1) to (b1) (t2) to (b5) (t3) to (b3) (t4) to (b4)  (t5) to (b2) (t6) to (b6);
\node[above] at (t2) {\tiny\alert{$\phantom{j}i\ \phantom{j}$}};\node[above] at (t5) {\tiny\alert{$j$}};\node[below, white] at (b4) {\tiny{$j$}};
\node at (1.5, .5){\tiny$\dots$}; \node at (5.5, .5){\tiny$\dots$};  \node at (3.5, .75){\tiny$\dots$}; 
}
\quad \text{ and } \quad 
e_{i,j} = \TikZ{[diagram]
\foreach\x/\y in {1/1, 2/2, 2.5/3, 4.5/4, 5/5, 6/6}{\node[v] (t\y) at (\x,1){};\node[v] (b\y) at (\x,0){};}
\draw (t1) to (b1)  (t3) to (b3) (t4) to (b4)(t6) to (b6)
		(t2) to [bend right=20] node[bT]{} (t5) (b5) to [bend right=20] node[D]{} (b2) ;
\node[above] at (t2) {\tiny\alert{$\phantom{j}i\ \phantom{j}$}};\node[above] at (t5) {\tiny\alert{$j$}};\node[below, white] at (b4) {\tiny{$j$}};
\node at (1.5, .5){\tiny$\dots$}; \node at (5.5, .5){\tiny$\dots$};  \node at (3.5, .5){\tiny$\dots$}; 
}\ .
$}}\pause 


The Brauer algebra $B_d(\delta) = B_d(\delta, 1)$ has Jucys-Murphy elements 
$$x_j =c+ \sum_{i=1}^{j-1} s_{i,j} - e_{i,j}, \quad c\in \CC,\ j=1, \dots, k,$$
that pairwise commute (Nazarov 1996).  \pause Define the degenerate affine version $\cB_d(\delta)$ by \\ \smallskip
\centerline{$\displaystyle
\cB_d(\delta) = \CC[y_1, \dots, y_d] \otimes B_d(\delta)/\<y_i\text{-$B_d$ relations}\>.
$} \smallskip
\pause 

\alert{Action on tensor space:} In the classical case, $y_i$ acts on $M \otimes V^{\otimes d}$ via the split Casimir as $\sum_{i=0}^{j-1} \gamma_{i,j}$. \pause Now we have the analogous tool for the $\fg = \fp(V)$ case! \pause \alert{Goal:} Define degenerate affine marked Brauer and cyclotomic Brauer algebras, and classify their representation theory. 
\end{frame}


\begin{frame}
\vspace{-2pt}
\hbox{\hspace{-.51in}
\includegraphics[width=5.15in]{groupphoto-text}}

\end{frame}




\end{document}
