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 \title{Dyck combinatorics in $p$-Kazhdan--Lusztig theory }
%\author{C. Bowman}
  
 \author{Chris Bowman\thanks{\href{mailto:chris.bowman-scargill@york.ac.uk}{chris.bowman-scargill@york.ac.uk}}\addressmark{1}, Maud De Visscher\addressmark{2},   Amit  Hazi\addressmark{3},  \and Catharina Stroppel\addressmark{4}}
% \address{ Mathematical Institute, Endenicher Allee 60, 53115 Bonn}
% \email{Stroppel@math.uni-bonn.de}

\address{\addressmark{1}Department of Mathematics, University of York \\ \addressmark{2}Department of Mathematics, City, University of London \\ \addressmark{3}School of Mathematics, University of Leeds \\ \addressmark{4}Mathematical Institute of the University of Bonn}
 
 
\abstract{We survey some recent advances in combinatorial modular representation theory in type $A$ through the lens of $p$-Kazhdan--Lusztig theory.}
 
 
\begin{document}
 \maketitle
 

\section{Introduction}


 






The diagrammatic Hecke category has provided the intuition and  tools necessary to cut through  the most famous conjectures of Lie theory: the Lusztig and Kazhdan--Lusztig positivity conjectures. 
These conjectures place the Kazhdan--Lusztig polynomials (associated to parabolic Coxeter systems) 
centre-stage in the (modular) representation theory of Lie theoretic objects.








































  Kazhdan--Lusztig polynomials encode a great deal of character-theoretic 
 and indeed cohomological information about cell modules.
 We further know that  Kazhdan--Lusztig polynomials often carry   information about the radical layers of   indecomposable   projective and cell modules.  
Given the almost ridiculous level of detail these polynomials encode, 
 it is natural to ask {\em``what are the limits to what $p$-Kazhdan--Lusztig combinatorics can tell us about the  structure of the Hecke category?"}


%The starting point of this  paper is to delve deep  into the Dyck/Temperley--Lieb combinatorics for $p$-Kazhdan--Lusztig polynomials, which was initiated in 
%    \cite{MR2918294,MR4323501,compan}.  There is a  wealth of extra, richer combinatorial information which can be encoded into the Dyck tilings underlying these
%    $p$-Kazhdan--Lusztig 
%     polynomials.  
%     Instead of looking only at the sets of Dyck tilings 
%     (which enumerate the $p$-Kazhdan--Lusztig 
%     polynomials) we look at the  relationships for passing 
%     between
%     these Dyck tilings.  
%     In fact, this ``meta-Kazhdan--Lusztig combinatorics"  
%      is sufficiently rich as to completely determine the full structure of our Hecke categories:
%       
% \begin{thmB}
%% Let $\Bbbk$ be a field. The ${\rm Ext}$-quiver of  
%The $\Bbbk$-algebra  $ \mathcal{H}_{m,n} $  admits a quadratic presentation as the path algebra of the ``Dyck quiver" $\mathscr{D}_{m,n}$ of \cref{quiverdefn}
% modulo ``Dyck-combinatorial relations" \eqref{rel1} to \eqref{adjacent}.
%If  $\Bbbk$ is a field,  then  the ${\rm Ext}$-quiver of  
%$\mathcal{H}_{m,n} 
%$  is isomorphic  to $\mathscr{D}_{m,n}$ 
%and  this gives a presentation of the algebra by quiver and relations.
%  \end{thmB}
%
% In a nutshell,  the power of Theorem B is that it allows us to understand not only the 
%{\em graded composition series} of standard and projective modules 
% (the purview of classical Kazhdan--Lusztig combinatorics) but the {\em explicit 
% extensions interrelating  these composition factors} 
% (in terms of      meta-Kazhdan--Lusztig combinatorics).
%   In essence, Theorem B  provides complete information about the structure of the anti-spherical Hecke categories 
%of $(S_{m+n},S_m \times S_n)$ for $m,n\in\NN$.  
% We reap some of the fruits of Theorem B  by providing an incredibly  elementary description of the  full submodule lattices of Verma modules:









%The truth of Lusztig's conjecture, which concerns the modular setting, was never questioned and the search for a proof was the backbone of 20th century Lie theory.  
%In 2012, Williamson found startling counterexamples to Lusztig's conjecture within the diagrammatic Hecke category and upended the status quo of modular representation theory.  
%More precisely, Williamson showed that the characteristic of the field must be exponentially enormous in order for the Kazhdan--Lusztig polynomials to control the representation theory of reductive groups.
%%However, this was only the beginning of the story.  
%The diagrammatic Hecke category provides the setting in which we can finally understand this chasm between representation theory over fields of positive characteristic and characteristic $0$.  
%This problem will undoubtedly be the core of Lie theory in the 21st century.
%
%
%
%
%
%
%
%
%
%The diagrammatic Hecke category  was introduced as
% an interface between Kazhdan--Lusztig combinatorics and  (modular) Lie theory.  This diagrammatic setting provided the 
%intuition  and    tools necessary  to resolve the Lusztig and Kazhdan--Lusztig positivity conjectures.
 
 


The  family of 
ordinary  Kazhdan--Lusztig polynomials which are
 combinatorially best understood   are those for 
maximal parabolics of finite symmetric groups $\mathfrak{S}_m \times \mathfrak{S}_n \leq \mathfrak{S}_{m+n}$.  
These polynomials can be calculated in terms of the  combinatorics 
of 
 Dyck tilings  \cite{MR646823}.  
       The starting point of this  project  was to extend this to the modular case by proving that the $p$-Kazhdan--Lusztig polynomials of $\mathfrak{S}_m \times \mathfrak{S}_n \leq \mathfrak{S}_{m+n}$ are entirely independent of $p\geq0$. 
       % to delve deeper  into this    Dyck  combinatorics to obtain more structural.  W
       We  also find that there is a  wealth of extra, richer combinatorial information which can be encoded into the Dyck tilings.  
     Instead of looking only at the sets of Dyck tilings 
     (which enumerate these  Kazhdan--Lusztig 
     polynomials) we look at the  relationships for passing 
     between
     these Dyck tilings.  
     In fact, this ``meta-Kazhdan--Lusztig combinatorics"  
      is sufficiently rich as to completely determine the full structure of our Hecke categories. 
 In this extended abstract, we discuss how this allows us to provide a complete combinatorial description of the   submodule lattices of the cell modules for these categories. 
 
 
  
%  These polynomials  admit inexplicably simple     
%        combinatorial formulae in terms of  Dyck paths            \cite{MR646823}. 
We also  proved in \cite{compan2} that  the Hecke categories of 
$\mathfrak{S}_m \times \mathfrak{S}_n \leq \mathfrak{S}_{m+n}$ 
  control  the structure  of parabolic Verma modules for Lie algebras \cite{MR2781018,MR888703,MR646823}; 
the representation category  of the general linear supergroups \cite{MR1937204};  
 arc algebras from categorified knot theory \cite{MR2881300};    
    walled Brauer algebras \cite{MR2813567};  
%    and  categories $\mathcal{O}$ for Grassmannians \cite{MR646823}.
    and the combinatorics of attracting cells for torus fixed points in Springer fibers \cite{MR2914857}.  
 This makes the cell modules of these categories  some of the most well-understood  
representations   in all of  non-semisimple Lie theory.

 
 
 
 
 
 
    
%    and  categories $\mathcal{O}$ for Grassmannians \cite{MR646823}.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
  
\section{Kazhdan--Lusztig polynomials}\label{reduced-defn}
 
  
  Let $(W, S_W)$ be a Coxeter system:  $W$ is the group generated by the finite set $S_W$ subject to the relations $(\csigma\ctau)^{m_{\csigma\ctau}} = 1$ for  
 $\csigma,\ctau\in S_W$, $ {m_{\csigma\ctau}}\in \NN\cup\{\infty\}$ 
 satisfying ${m_{\csigma\ctau}}= {m_{\ctau\csigma}}$, and ${m_{\csigma\ctau}}=1$ if and only if  $\csigma= \ctau$.  
   Let $\ell : W \to \mathbb{N}$  be the corresponding length function. 
  Consider $S_P \subseteq S_W$ a  subset and $(P, S_P)$
 its corresponding Coxeter system. We say that $P$ is the parabolic subgroup corresponding to  $S_P\subseteq S_W$.
Let  $^ PW \subseteq W$ denote a set of minimal coset representatives in $P\backslash W$. 
 For $\w=\sigma_1\sigma_2\cdots  \sigma_\ell$ an expression, we define a subword  to be a sequence
$\underline{t}=(t_1,t_2,\dots ,t_\ell)\in\{0,1\}^\ell$ and   set 
 $\w^{\underline{t}}  :=\sigma_1^{t_1}\sigma_2^{t_2}\cdots \sigma_\ell^{t_\ell}$. % and we emphasise that $s_i^0=1  \in W$.   
  We let $\leq $ denote the strong  Bruhat order on $^PW$: namely $y\leq w$ if for some  
reduced expression $\w$ there exists   a subword ${\underline{t}}$ and a reduced expression  $\y$ such that $\w^{\underline{t}}=\y$.   We denote the Hasse diagram of this poset  by $\mathcal{G}_{(W,P)}$  {and we refer to it as the {\em  Bruhat graph} of the pair $(W,P)$.}  Explicitly, the vertices of  $\mathcal{G}_{(W,P)}$ are labelled by the elements of $^PW$ and for $\lambda \in {^PW}$ we have a directed edge  $\lambda \rightarrow \lambda s_i$ if  $\lambda < \lambda s_i \in {^PW}$ for some $s_i\in S_W$.
We denote by $\varnothing$ (for the empty word in the generators) the minimal coset representative for the identity coset $P$. 
 



































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%
%\draw[very thick, rounded corners] (-10.8+1.8,2.5-1) rectangle (-7.2+1.8,3.5-1); 
%
%\draw[very thick, rounded corners] (-4.8+1.8,2.5-1) rectangle (-3.2+1.8,3.5-1); 




 
 
\draw[very thick]( -8.5+1.8,3-1)--(-4+1.8,3-1); 
 
\draw[very thick, fill=darkgreen] ( -7+1.8,3-1) coordinate circle (7pt);  

\draw[very thick, fill=magenta] ( -5.5+1.8,3-1) coordinate circle (7pt); 
 
\draw[very thick, fill=cyan] ( -4+1.8,3-1) coordinate circle (7pt); 

%\draw[very thick, rounded corners] (-7.5+1.8,2.5-1) rectangle (-7.2+1.8,3.5-1); 
%
%\draw[very thick, rounded corners] (-4.8+1.8,2.5-1) rectangle (-3.2+1.8,3.5-1); 
 \draw[very thick, fill=orange] ( -8.5+1.8,3-1) coordinate circle (7pt); 


\draw[very thick, rounded corners] (-6.5+1.8,2.5-1) rectangle (-9+1.8,3.5-1); 
\draw[very thick, rounded corners] (-4.5+1.8,2.5-1) rectangle (-3.5+1.8,3.5-1); 


\end{tikzpicture}
$$

\caption{The  graph   $\mathcal{G}_{(W,P)}$ for  $(W,P)=(\mathfrak{S}_4,\mathfrak{S}_2 \times \mathfrak{S}_2 )$ and $(\mathfrak{S}_5, \mathfrak{S}_2 \times \mathfrak{S}_3 )$ respectively. 
%(We haven't drawn the direction on the edges but all arrows are pointing upwards).
 } %$A_4 \setminus A_2 \times A_1$. }
\label{Bruhatexample-KL}
\end{figure} 
 
We define  the {\em  extended Bruhat graph} $\widehat{\mathcal{G}}_{(W,P)}$  to be the directed graph having the same set of vertices as $\mathcal{G}_{(W,P)}$ but replacing each edge in $\mathcal{G}_{(W,P)}$  between $\lambda$ and $\lambda s_i$ for $\lambda < \lambda s_i$ by four  
 ``up" and ``down" directed edges  
\begin{align}\label{begin!}
  \lambda \xrightarrow{ \ i \ }\lambda s_i ,
   \quad
\lambda \xrightarrow{ \ i \ }\lambda , 
\quad
      \lambda s_i \xrightarrow{ \ i \ }\lambda 
 \quad
  \lambda s_i  \xrightarrow{ \ i \ }\lambda s_i,
 \end{align}
which we denote $ {\sf U}_i^1$, ${\sf U}_i^0$,    ${\sf D}_i^1$, ${\sf D}_i^0$    respectively.  
  We 
assign a degree to each edge in $\widehat{\mathcal{G}}_{(W,P)}$ by setting 
\[{\rm deg}(\lambda \xrightarrow{ \ i \ } \lambda s_i) = {\rm deg}(\lambda s_i  \xrightarrow{ \ i \ } \lambda) = 0  
\qquad {\rm deg}(\lambda \xrightarrow{ \ i \ } \lambda) = \left\{ \begin{array}{ll} 1 & \mbox{if $\lambda s_i > \lambda$}\\ -1 & \mbox{if $\lambda s_i < \lambda$}\end{array}\right.\]
%{\color{magenta} A walk on the usual Bruhat graph $\mathcal{G}_{(W,P)}$ can be thought of as a path in the extended Bruhat graph $\widehat{\mathcal{G}}_{(W,P)}$ . 
 Given   a path   (or ``Bruhat stroll") on $\widehat{\mathcal{G}}_{(W,P)}$
\[\SSTT \, : \, \lambda_1 \xrightarrow{i_1} \lambda_2 \xrightarrow{i_2} \lambda_3 \xrightarrow{i_3} \ldots \xrightarrow{i_{k-1}} \lambda_k,\]
we say that the {\em  degree}  ${\rm deg}(\SSTT)$ is the sum of the degrees of each edge in $\SSTT$. 
(The degree is also sometimes known as the ``Deodhar defect".)    
We also define the {\sf weight of $\SSTT$}, denoted by $w(\SSTT)$ to be the expression 
\[w(\SSTT) :=  {s_{i_1}s_{i_2}s_{i_3} \ldots s_{i_{k-1}}}.\]
%We write $\SSTT_\la \otimes \csigma$ for the path of weight 
%$ {s_{i_1}s_{i_2}s_{i_3} \ldots s_{i_{k-1}}}\csigma$.  
Given $\la \in {^PW}$, we let $\Path (\la)$ denote the set of all paths from  $\varnothing$ and ending at $\la$ in the extended Bruhat graph.  
    
  





























 
 
 
 
 
 
 
 
 
 
 
 
  
\begin{defn}
 We say that a path $\SSTT\in \Path (\mu)$  is {\em  reduced} if it is a 
path of shortest possible length from $\varnothing$ to $\mu$.  
  \end{defn}
      
 
 
    
    
 
Throughout the paper we will fix one reduced path, $\SSTT^\mu \in \Path (\mu)$, for each $\mu \in {^PW}$.  For a fixed $\la$, we denote the set of all  paths $\SSTT\in \Path(\la)$ with $w(\SSTT) = \SSTT^\mu$  by 
$\Path (\la,\SSTT^\mu)$.  
 
Examples are given in Figure 2.   
    
 
    
     
   \begin{defn}\label{troll}
 Given $(W,P)$ a parabolic Coxeter system, we define the {\em  matrix of light-leaves polynomials}
 \[
\Delta ^{(W,P)}:=( \Delta _{\la,\mu}(q))_{\la,\mu\in {{^P}W}}
\qquad \qquad
\Delta _{\la,\mu}(q)=\sum_{\SSTS \in {\rm Path}(\la,\SSTT^\mu)}\!\!\!\!\!\! q^{\deg(\SSTS)}
\]
which is a  (square) lower uni-triangular matrix.  
This matrix can be factorised {\em  uniquely} as a product of 
lower uni-triangular matrices 
\[N^{(W,P)}:=(n_{\la,\nu}(q))_{\la,\nu\in {{^P}W}}
\qquad 
B^{(W,P)}:=(b_{\nu,\mu}(q))_{\nu,\mu\in {{^P}W}} 
\]
such that $n_{\la,\nu}(q)\in q\ZZ[q]$ 
for $\la \neq \nu$ 
and 
$b_{\nu,\mu}(q)\in  \ZZ[q+q^{-1}]$. 
The polynomials $n_{\la,\nu}(q)$ are the anti-spherical  Kazhdan--Lusztig polynomials of  $(W,P)$. %We set 
%$b_{\la,\la}(q)=1=n_{\la,\la}$.  
%For $\la\neq \mu$, we recursively define polynomials
%\[
%b_{\la,\mu}(q)\in \ZZ [q+q^{-1}] \qquad
%n_{\la,\mu}(q)\in q\ZZ [q]
%\]
%  by induction on the Bruhat order $\leq$ as follows
%\begin{equation}\label{kl-pol-d}
%b_{\la,\mu}(q) +n_{\lambda,\mu}(q) =
%\!\!\! \sum_{\SSTS \in {\rm Path}(\la,\SSTT^\mu)}\!\!\!\!\!\! q^{\deg(\SSTS)} -
% \!\!\sum_{
%\begin{subarray}c
%\la <  \nu <  \mu\end{subarray}
%}   \!\!\!   n_{\lambda,\nu}(q)
%b_{\nu,\mu}(q).  
%\end{equation}
%The polynomials $n_{\la,\mu}(q)$ are called the {\em Kazhdan--Lusztig polynomials} associated to $\la,\mu$.
   \end{defn}
 


\begin{figure} [ht!]
$$\qquad\quad
\begin{tikzpicture} [yscale=0.425,xscale=-0.425]
 
\path[magenta, line width=3] (2,0.5)--(2,3) coordinate (hi);

\path[cyan, line width=3] (hi)--(6,5.5) coordinate (hi2);
\path[darkgreen, line width=3] (hi)--(-2,5.5) coordinate (hi3);
\path[cyan, line width=3] (hi3)--(2,8) coordinate (hi4);
\path[darkgreen, line width=3] (hi2)--(2,8) coordinate (hi4);
\path[orange, line width=3] (hi3)--(-6,8) coordinate (hi5);
\path[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);
\path[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);
\path[magenta, line width=3] (hi4)--(6,10.5) coordinate (hi7);


\path[magenta, line width=3] (hi6)--(2,13) coordinate (hi8);
\path[orange, line width=3] (hi7)--(2,13) coordinate (hi8);
\path[darkgreen, line width=3] (hi8)--(2,15.5) ; 

 

\path (0,0) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;





  
\draw[magenta, line width=3] (2,0.5)--(2,3) coordinate (hi);

\path (0,0) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;



\path (0,2.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


















  
\draw[darkgreen, line width=3] (hi)--(-2,5.5) coordinate (hi3);
 
\path (0,2.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (-4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;









  
\draw[cyan, line width=3] (hi)--(6,5.5) coordinate (hi2);

 

\path (0,2.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;











  

\draw[cyan, line width=3] (hi3)--(2,8) coordinate (hi4);
\draw[darkgreen, line width=3] (hi2)--(2,8) coordinate (hi4);


\path (-4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;


\path (0,7.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;






















  

\draw[orange, line width=3] (hi3)--(-6,8) coordinate (hi5);
%\draw[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);



\path (-4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;



\path (-8,7.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(90:0.175) node  {  $   \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;
































  

\draw[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);
\draw[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);



\path (-8,7.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(90:0.175) node  {  $   \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;





\path (0,7.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;



\path (-4,10) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;





  

%\draw[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);
 \draw[magenta, line width=3] (hi4)--(6,10.5) coordinate (hi7);



\path (-8,7.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(90:0.175) node  {  $   \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;





\path (0,7.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;







 







\path (4,10) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;





















  


 
\draw[magenta, line width=3] (hi6)--(2,13) coordinate (hi8);
\draw[orange, line width=3] (hi7)--(2,13) coordinate (hi8);
%\draw[darkgreen, line width=3] (hi8)--(2,15) ; 



\path (-4,10) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;

\path (4,10) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;




\path (0,12.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;




  


\draw[darkgreen, line width=3] (hi8)--(2,15) ; 

\path (0,12.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;







\path (0,15) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;




  


%\draw[line width=3  ]  (2,0.5)--++(90:2.5) --(-2,5.5)--(-6,8)--(2,13)--++(90:2.5); 
      


 
\fill(hi)circle (8pt);
   
    
  
     \draw[darkgreen, line width=3] (hi8)--++(90:2.5)  ;  
  \path (hi8)--++(90:2.5) coordinate (ghfj)
;  \fill  (ghfj)
circle (8pt);

      \draw[line width=3,  , ] (2,0.5)--++(90:2.5)--(-2,5.5)
 --(-6,8) --(-2,10.5)--(2,13)--++(90:2.5)
 coordinate(hi);


\draw[fill=black] (hi)circle  (22pt); 

\foreach \i in {2,3,...,8}
{\fill(hi\i)circle (8pt);	}


\draw  (hi)
node { \color{white}$  \alpha$}; 


\fill(2,0.5)circle (8pt);
  \fill(2,3)circle (8pt);

 
 \path (2,15.5) circle (22pt);
\end{tikzpicture}\qquad
\quad\begin{tikzpicture} [yscale=0.425,xscale=-0.425]
 
\path[magenta, line width=3] (2,0.5)--(2,3) coordinate (hi);

\path[cyan, line width=3] (hi)--(6,5.5) coordinate (hi2);
\path[darkgreen, line width=3] (hi)--(-2,5.5) coordinate (hi3);
\path[cyan, line width=3] (hi3)--(2,8) coordinate (hi4);
\path[darkgreen, line width=3] (hi2)--(2,8) coordinate (hi4);
\path[orange, line width=3] (hi3)--(-6,8) coordinate (hi5);
\path[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);
\path[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);
\path[magenta, line width=3] (hi4)--(6,10.5) coordinate (hi7);


\path[magenta, line width=3] (hi6)--(2,13) coordinate (hi8);
\path[orange, line width=3] (hi7)--(2,13) coordinate (hi8);
\path[darkgreen, line width=3] (hi8)--(2,15.5) ; 

 

\path (0,0) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;





  
\draw[magenta, line width=3] (2,0.5)--(2,3) coordinate (hi);

\path (0,0) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;



\path (0,2.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


















  
\draw[darkgreen, line width=3] (hi)--(-2,5.5) coordinate (hi3);
 
\path (0,2.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (-4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;









  
\draw[cyan, line width=3] (hi)--(6,5.5) coordinate (hi2);

 

\path (0,2.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;











  

\draw[cyan, line width=3] (hi3)--(2,8) coordinate (hi4);
\draw[darkgreen, line width=3] (hi2)--(2,8) coordinate (hi4);


\path (-4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;


\path (0,7.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;






















  

\draw[orange, line width=3] (hi3)--(-6,8) coordinate (hi5);
%\draw[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);



\path (-4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;


\path (4,5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;



\path (-8,7.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(90:0.175) node  {  $   \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;
































  

\draw[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);
\draw[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);



\path (-8,7.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(90:0.175) node  {  $   \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;





\path (0,7.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;



\path (-4,10) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;





  

%\draw[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);
 \draw[magenta, line width=3] (hi4)--(6,10.5) coordinate (hi7);



\path (-8,7.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(90:0.175) node  {  $   \down   $} ;
 %\path(a5) --++(-90:0.175) node  {  $  \up   $} ;





\path (0,7.5) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down    $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;







 







\path (4,10) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;





















  


 
\draw[magenta, line width=3] (hi6)--(2,13) coordinate (hi8);
\draw[orange, line width=3] (hi7)--(2,13) coordinate (hi8);
%\draw[darkgreen, line width=3] (hi8)--(2,15) ; 



\path (-4,10) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down  $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;

\path (4,10) coordinate (origin); 

 %\path(a1) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up  $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;




\path (0,12.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;




  


\draw[darkgreen, line width=3] (hi8)--(2,15) ; 

\path (0,12.5) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(90:0.175) node  {  $  \down $} ;
 %\path(a3) --++(-90:0.175) node  {  $  \up   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;







\path (0,15) coordinate (origin); 

 %\path(a1) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a2) --++(-90:0.175) node  {  $  \up   $} ;
 %\path(a3) --++(90:0.175) node  {  $  \down   $} ;

 %\path(a4) --++(90:0.175) node  {  $  \down   $} ;
 %\path(a5) --++(90:0.175) node  {  $  \down  $} ;




  


%\draw[line width=3  ]  (2,0.5)--++(90:2.5) --(-2,5.5)--(-6,8)--(2,13)--++(90:2.5); 
  
\draw[line width=3  ,,]  (2,0.5)--++(90:2.5) coordinate(hi);
    
\draw[line width=3,  , ] (2,0.5)--++(90:2.5)--(-2,5.5)
coordinate(hi);
    
\draw[line width=3  , ] (hi) to
 [out=180,in=-120] 
(-4,6.75) 
 to [out=60,in=90] (-2,5.5) 
;
   
 \draw[line width=3 , , ] (-2,5.5) --(2,8); 
    



  \draw[line width=3 , ]   (2,8)  
to [out=0,in=-60] 
(4,9.25) 
  to [out=120,in=90] (2,8) 
; 
\fill(hi)circle (8pt);
   
  \draw[line width=3 , ]   (2,8)--(6,5.5)  
  coordinate (hi); 
  
  
     \draw[darkgreen, line width=3] (hi8)--++(90:2.5)  ;\foreach \i in {2,3,...,8}
{\fill(hi\i)circle (8pt);	}
 
  \path (hi8)--++(90:2.5) coordinate (ghfj)
;  \fill  (ghfj)
circle (8pt);

   

\draw[fill=black] (hi)circle  (22pt); 



\draw  (hi)
node { \color{white}$  \beta$}; 


\fill(2,0.5)circle (8pt);
  \fill(2,3)circle (8pt);

 
 \path (2,15.5) circle (22pt);
\end{tikzpicture}
$$  
\caption{On the left we depict a path $\SSTT^\alpha$ and on the right we depict 
the unique element    $\SSTS \in \Path(\beta,\SSTT^\alpha)$ for $\alpha=\color{magenta}s_2\color{darkgreen}s_3\color{orange}s_4  \color{cyan}s_1 \color{magenta}s_2\color{darkgreen}s_3$ and 
$\beta=\color{magenta}s_2   \color{cyan}s_1  $.  
These are paths  on $\widehat{\mathcal{G}}_{ {(\mathfrak{S}_5, \mathfrak{S}_2\times \mathfrak{S}_3)} } $   (also known as ``Bruhat strolls") but we depict only the edges in $\mathcal{G}_{ {(\mathfrak{S}_5, \mathfrak{S}_2\times \mathfrak{S}_3)} }$ (for readability).
  }
\label{pppspspspsps}
\end{figure}

     
%We can reformulate the above in terms of matrix multiplication.  
%We define the {\sf matrix of light-leaves polynomials}
%\[
%\Delta^\Bbbk^{(W,P)}:=( \Delta^\Bbbk_{\la,\mu}(q))_{\la,\mu\in {{^P}W}}
%\qquad \qquad
%\Delta^\Bbbk_{\la,\mu}(q)=\sum_{\SSTS \in {\rm Path}(\la,\SSTT^\mu)}\!\!\!\!\!\! q^{\deg(\SSTS)}
%\]
%to be the (square) lower uni-triangular matrix whose entries record the degrees of paths of a given weight and shape.   
%This matrix can be factorised {\em  uniquely} as a product 
%$\Delta^\Bbbk^{(W,P)}= N^{(W,P)} \times B^{(W,P)}$ of  
%lower uni-triangular matrices 
%\[N^{(W,P)}:=(n_{\la,\nu}(q))_{\la,\mu\in {{^P}W}}
%\qquad 
%B^{(W,P)}:=(b_{\nu,\mu}(q))_{\nu,\mu\in {{^P}W}}
%\]
%such that $n_{\la,\nu}(q)\in q\ZZ[q]$ 
%for $\la \neq \nu$ 
%and 
%$b_{\nu,\mu}(q)\in  \ZZ[q+q^{-1}]$. 


\begin{eg}
The matrix $\Delta^\Bbbk$ in type $(\mathfrak{S}_4,\mathfrak{S}_2\times \mathfrak{S}_2)$ is depicted below. \[
%\setlength{\arraycolsep}{8pt}
 \begin{array}{r|cccccc}
\Delta^\Bbbk  
&  \color{magenta}s_2\color{cyan}s_1  \color{darkgreen}s_3 \color{magenta}s_2
    &  \color{magenta}s_2\color{cyan}s_1  \color{darkgreen}s_3 
&  \color{magenta}s_2 \color{cyan}s_1 
    &\color{magenta}s_2 \color{darkgreen}s_3   
&  \phantom{s_2}\color{magenta}s_2 \phantom{s_2}  
& \phantom{s_2} \varnothing \phantom{s_2}      
         \\ \hline
 \color{magenta}s_2\color{cyan}s_1  \color{darkgreen}s_3 \color{magenta}s_2 
   & 1 & \cdot  & \cdot  & \cdot  & \cdot  & \cdot  \\
 	 \color{magenta}s_2\color{cyan}s_1  \color{darkgreen}s_3 			 & q & 1 & \cdot  & \cdot  & \cdot  & \cdot  \\
 \color{magenta}s_2\color{cyan}s_1  & \cdot  & q & 1 & \cdot  & \cdot  & \cdot  \\
 \color{magenta}s_2  \color{darkgreen}s_3    & \cdot  & q & \cdot  & 1 & \cdot  & \cdot  \\
  \color{magenta}s_2  & q & q^2 & q & q & 1 & \cdot  \\
\varnothing   & q^2 & \cdot  & \cdot  & \cdot  & q & 1
\end{array}\]
  The factorisation of this matrix is trivial, with $N=\Delta^\Bbbk$ and $B={\rm Id}_{6\times 6}$ the identity matrix.
 \end{eg}

 The Hecke category (over the complex field) gives a categorification of this matrix factorisation.  
 
 
 

\section{Hecke categories and $p$-Kazhdan--Lusztig polynomials}
Hecke categories provide the interface between  Lie theory and 
Kazhdan--Lusztig theory.  
We begin by lifting the ``folded paths" of the previous section to provide (what will be) a basis of the ${\rm Hom}$-spaces of the Hecke category.

In this section, we will only explicitly  discuss the generators and relations for $\mathscr{H}_{(W,P)}$, the category algebra of the Hecke category,  when 
 $W=\mathfrak{S}_{n+m}$ is a finite symmetric group and $P$ is a maximal parabolic $P=\mathfrak{S}_m \times \mathfrak{S}_n$, as this simplifies the definitions  considerably, whilst still illustrating the important points of the general case. 
 We define the {\em  Soergel generators} to be the framed graphs 
$$
{\sf 1}_{\emptyset } =
\begin{minipage}{1.5cm} \begin{tikzpicture}[scale=1]
\draw[densely dotted,rounded corners](-0.5cm,-0.6cm)  rectangle (0.5cm,0.6cm);
\clip(0,0) circle (0.6cm);
%\draw[line width=0.06cm, magenta](0,-1)--(0,+1);
\end{tikzpicture}
\end{minipage}
\quad
{\sf 1}_{\csigma } = 
\begin{minipage}{1.5cm} \begin{tikzpicture}[scale=1]
\draw[densely dotted,rounded corners](-0.5cm,-0.6cm)  rectangle (0.5cm,0.6cm);
\clip(0,0) circle (0.6cm);
\draw[line width=0.06cm, magenta](0,-1)--(0,+1);
\end{tikzpicture}
\end{minipage}
\quad
   {\sf spot}_\csigma^\emptyset
  =
  \begin{minipage}{1.5cm}  
  \begin{tikzpicture}[scale=1]
\draw[densely dotted,rounded corners](-0.5cm,-0.6cm)  rectangle (0.5cm,0.6cm);

\clip(0,0) circle (0.6cm);
\draw[line width=0.06cm, magenta](0,-1)--(0,+0);
\fill[magenta] (0,0)  circle (5pt);
\end{tikzpicture}\end{minipage}
\quad 
{\sf fork}_{\csigma\csigma}^{\csigma}= 
   \begin{minipage}{2cm}  
 \begin{tikzpicture}[scale=1]
 \draw[densely dotted,rounded corners](-0.75cm,-0.5cm)
  rectangle (0.75cm,0.5cm);
  \clip (-0.75cm,-0.5cm)
  rectangle (0.75cm,0.5cm);
  \draw[line width=0.08cm, magenta](0,0)to [out=-30, in =90] (10pt,-15pt);
 \draw[line width=0.08cm, magenta](0,0)to [out=-150, in =90] (-10pt,-15pt);
 \draw[line width=0.08cm, magenta](0,0)--++(90:1);
 \end{tikzpicture}
\end{minipage}
\quad
{\sf braid}_{\csigma\ctau}^{\ctau\csigma}= 
   \begin{minipage}{1cm}  
 \begin{tikzpicture}[scale=1.1]
 \draw[densely dotted,rounded corners](-0.5cm,-0.5cm)  rectangle (0.5cm,0.5cm);
\clip(-0.5cm,-0.5cm)  rectangle (0.5cm,0.5cm);
%\clip(0,0) circle (15pt);
 \draw[line width=0.08cm, magenta] (0,0) to [out=45, in =-90] (10pt,0.5cm);
 \draw[line width=0.08cm, magenta] (0,0) to [out=-135, in =90] (-10pt,-0.5cm);
 
 
 
 
  \draw[line width=0.08cm, cyan] (0,0) to [out=-45, in =90] (10pt,-0.5cm);
 \draw[line width=0.08cm, cyan] (0,0) to [out=135, in =-90] (-10pt,0.5cm);
    \end{tikzpicture}\end{minipage} 
    $$  
associated to any pair $\csigma, \ctau\in S_W$ with $m_{\csigma \ctau}=2$.
We define the northern/southern reading word of any diagram obtained from
 horizontal and vertical concatenation of  
  Soergel generators to  be the word in the alphabet $S_W  $ which records the colours along   the northern/southern edge of the frame respectively.  
 We let $\otimes$ to be horizontal concatenation of diagrams, the  algebra multiplication $\circ$ will be given by vertical concatenation in the usual manner for diagram algebras.
We let $\ast$ denote the anti-involution which flips a diagram  through the horizontal axis.  



 \begin{defn}\label{upanddown}
 We   define up and down operators on diagrams as follows
  \begin{itemize}[leftmargin=*]
\item    
%For  $D$ an arbitrary diagram, we define
Suppose that $D $ has northern colour sequence $\SSTT^\la  $ with   
$\la\csigma  >\la$.  We define  $$
\qquad {\sf U}^1_\csigma(D)=\begin{minipage}{1.85cm}
\begin{tikzpicture}[scale=1.5]
\draw[densely dotted, rounded corners] (-0.5,0) rectangle (0.75,0.75) node [midway] {$ D$} ;
%\draw[magenta,line width=0.08cm](0,0)--++(90:0.75);
% \end{tikzpicture}\end{minipage}
%\otimes \begin{minipage}{0.75cm}\begin{tikzpicture}[scale=1.5]
\draw[densely dotted, rounded corners] (-0.25+1,0) rectangle (0.25+1,0.75);
\draw[magenta,line width=0.08cm](0+1,0)--++(90:0.75);
 \end{tikzpicture}\end{minipage}
 \qquad 
 \qquad  \qquad  \qquad 
 {\sf U}^0_\csigma(D)=
\begin{minipage}{1.85cm}\begin{tikzpicture}[scale=1.5]
\draw[densely dotted, rounded corners] (-0.5,0) rectangle (0.75,0.75) node [midway] {$D$} ;
%\draw[magenta,line width=0.08cm](0,0)--++(90:0.75);
% \end{tikzpicture}\end{minipage}
% \otimes 
%\begin{minipage}{0.75cm}\begin{tikzpicture}[scale=1.5]
\draw[densely dotted, rounded corners] (-0.25+1,0) rectangle (0.25+1,0.75);
\draw[magenta,line width=0.08cm](0+1,0)--++(90:0.3525) coordinate (hi);
\draw[fill=magenta,magenta] (hi) circle (3pt);
 \end{tikzpicture}\end{minipage} \qquad  \qquad  \qquad 
$$
 
\item   Now suppose that $D $ has northern colour sequence $   \SSTT^\la \otimes    \csigma $ with   
$\la\csigma >\la$.
%We let  $\y  \in  $ be obtained by deleting $\x {\color{magenta}[r,c]}=k$ then we let  
% $\y ^{-1}(j)=\x ^{-1}(j)$ for $ 1\leq j <k$ and 
%  $\y ^{-1}(j-1)=\x ^{-1}(j)$ for $ k<j \leq a $. 
%We  let $\y  \otimes \csigma  \in \Path(\alpha)$ be defined by $\y {\color{magenta}[r,c]}=a$ and $(\y \otimes \csigma)[x,y] =\y [x,y]$ otherwise. 
We define 
$$
 {\sf D}_\csigma^0(D)= 
\begin{minipage}{1.85cm}\begin{tikzpicture}[scale=1.5]
\draw[densely dotted, rounded corners] (-1,0) rectangle (0.75,0.75) node [midway] { $D $} ;
 
\draw[densely dotted, rounded corners] (-1,0.75) rectangle (0.25,1.5)  node [midway] {${\sf 1}_{\SSTT^\la}$}; 
%\draw[densely dotted, rounded corners] (-1,-0.75) rectangle (0.75,0)  node [midway] {$D$}; 
;\draw[densely dotted, rounded corners] (1.25,0.75) rectangle (0.25,1.5)   ;
 \draw[densely dotted, rounded corners] (0.75,-0 ) rectangle (1.25,0.75);
\draw[magenta,line width=0.08cm](1,-0)--++(90:0.75) to [out=90,in=-10] (0.5,1.1);

%%SPOT ME 
\draw[magenta,line width=0.08cm](0.5,0.75)--++(90:0.75) ;
 \end{tikzpicture}\end{minipage}
 \qquad 
 \qquad 
 \qquad\quad
 {\sf D}_\csigma^1(D)= 
 \begin{minipage}{3.55cm}\begin{tikzpicture}[scale=1.5]
\draw[densely dotted, rounded corners] (-1,0) rectangle (0.75,0.75) node [midway] { $D$} ;
 
\draw[densely dotted, rounded corners] (-1,0.75) rectangle (0.25,1.5)  node [midway] {${\sf 1}_{\SSTT^\la }$}; %\draw[densely dotted, rounded corners] (-1,0) rectangle (0.75,0.75) node [midway] {${\sf braid}_{\SSTT^\la}^{\SSTT^{ \la - \csigma}\otimes\SSTT^{ \csigma}}$} ;
% 
%\draw[densely dotted, rounded corners] (-1,0.75) rectangle (0.25,1.5)  node [midway] {${\sf 1}_{\SSTT^{\lambda-\csigma}}$}; 
%\draw[densely dotted, rounded corners] (-1,-0.75) rectangle (0.75,0)  node [midway] {$D$};  
;\draw[densely dotted, rounded corners] (1.25,0.75) rectangle (0.25,1.5)   ;
 \draw[densely dotted, rounded corners] (0.75,-0 ) rectangle (1.25,0.75);
\draw[magenta,line width=0.08cm](1,-0)--++(90:0.75) to [out=90,in=-10] (0.5,1.02);
% \draw[densely dotted, rounded corners] (0.75,-0.75) rectangle (1.25,0.75);
%\draw[magenta,line width=0.08cm](1,-0.75)--++(90:1.5) to [out=90,in=-10] (0.5,1.1);
 
\draw[magenta,line width=0.08cm](0.5,0.75)--++(90:0.55) coordinate(hi);
\draw[fill=magenta,magenta] (hi) circle (3pt);
 \end{tikzpicture}\end{minipage}.\qquad$$

\end{itemize}
We do not emphasise the braids in our construction/notation since
 it will not matter if we pre- or post-multiply (at any stage of this construction) with a braid generator.  \end{defn} 

\begin{defn}
For  $\SSTS\in \Path(\la)$  we  construct a Soergel diagram by performing the up and down operators of \cref{upanddown} as we encounter 
any of the four up/down steps  in the path.  We denote the resulting diagram by $c_\SSTS$. The Soergel diagram corresponding to the path on the right hand side of Figure 2 is given below.
\end{defn}



$$  \begin{tikzpicture}[scale=1.5]
\draw (-0.65,0.325) node {$c_\SSTS=$};
\draw[densely dotted, rounded corners,fill=white] (-0.25,0) rectangle (0.25,0.75) ;
 \draw[magenta,line width=0.08cm](0,0)--++(90:0.75)  ;

\draw(0,0) node [below] {$ \bf \color{magenta} {\sf U}^1$};

 

\draw[densely dotted, rounded corners,fill=white] (0.25,0) rectangle (.5+.25,0.75) ;
 \draw[magenta,line width=0.08cm](0,0)--++(90:0.75)  ;
 \draw[darkgreen,line width=0.08cm](0.5,0)--++(90:0.75)  ;
 \draw(0,0) node [below] {$ \bf \color{magenta} {\sf U}^1$};
\draw(0.5,0) node [below] {$ \bf \color{darkgreen} {\sf U}^1$};










 

\draw[densely dotted, rounded corners,fill=white] (-0.25,0) rectangle (.75,0.75) ;
\draw[densely dotted, rounded corners,fill=white] (.75,0) rectangle (1.25 ,0.75) ;
 \draw[magenta,line width=0.08cm](0,0)--++(90:0.75)  ;
 \draw[darkgreen,line width=0.08cm](0.5,0)--++(90:0.75)  ;
 \draw[orange,line width=0.08cm](1,0)--++(90:0.4) coordinate(X)  ;
\fill[orange](X) circle (3.5pt);

  \draw(1,0) node [below] {$ \bf  \color{orange}\bf  {\sf U}^0$};
 

\draw[densely dotted, rounded corners,fill=white] (-0.25,0) rectangle (.75+.5,0.75) ;
\draw[densely dotted, rounded corners,fill=white] (.75+.5,0) rectangle (1.25+.5 ,0.75) ;
 \draw[magenta,line width=0.08cm](0,0)--++(90:0.75)  ;
 \draw[darkgreen,line width=0.08cm](0.5,0)--++(90:0.75)  ;
 \draw[orange,line width=0.08cm](1,0)--++(90:0.4) coordinate(X)  ;
\fill[orange](X) circle (3.5pt);
 \draw[cyan,line width=0.08cm](01.5,0)--++(90:0.75)  ;


 
%{\scalefont{1.5}$\SSTP=
%  \color{magenta}\bf  {\sf U}^1
% \color{darkgreen}\bf  {\sf U}^1
%  \color{orange}\bf  {\sf U}^0
%   \color{cyan}\bf  {\sf U}^1\color{magenta}\bf  {\sf U}^0 \color{darkgreen}\bf D^1 $};
 \draw(0,0) node [below] {$ \bf \color{magenta} {\sf U}^1$};
\draw(0.5,0) node [below] {$ \bf \color{darkgreen} {\sf U}^1$};
\draw(1,0) node [below] {$ \bf  \color{orange}\bf  {\sf U}^0$};
\draw(1.5,0) node [below] {$ \bf  \color{cyan}\bf  {\sf U}^1$};


 

\draw[densely dotted, rounded corners,fill=white] (-0.25,0) rectangle (.75+.5*2,0.75) ;
\draw[densely dotted, rounded corners,fill=white] (.75+.5*2,0) rectangle (1.25+.5*2 ,0.75) ;
 \draw[magenta,line width=0.08cm](0,0)--++(90:0.75)  ;
 \draw[darkgreen,line width=0.08cm](0.5,0)--++(90:0.75)  ;
 \draw[orange,line width=0.08cm](1,0)--++(90:0.4) coordinate(X)  ;
\fill[orange](X) circle (3.5pt);
 \draw[cyan,line width=0.08cm](01.5,0)--++(90:0.75)  ;
 
  \draw[magenta,line width=0.08cm](2,0)--++(90:0.4) coordinate(X)  ;
\fill[magenta](X) circle (3.5pt);
 
 
   
 \draw(0,0) node [below] {$ \bf \color{magenta} {\sf U}^1$};
\draw(0.5,0) node [below] {$ \bf \color{darkgreen} {\sf U}^1$};
\draw(1,0) node [below] {$ \bf  \color{orange}\bf  {\sf U}^0$};
\draw(2,0) node [below] {$ \bf  \color{magenta}\bf  {\sf U}^0$};
 
  
\draw[densely dotted, rounded corners,fill=white] (-0.25,1.5) rectangle (.75+.5*4,0.75) ;
\draw[densely dotted, rounded corners,fill=white] (-0.25,0) rectangle (.75+.5*3,0.75) ;
\draw[densely dotted, rounded corners,fill=white] (.75+.5*3,0) rectangle (1.25+.5*3 ,0.75) ;
 \draw[magenta,line width=0.08cm](0,0)--++(90:1.5)  ;
 \draw[darkgreen,line width=0.08cm](0.5,0)--++(90:1.3) coordinate(X)  ;
\fill[darkgreen](X) circle (3.5pt);


 \draw[orange,line width=0.08cm](1,0)--++(90:0.4) coordinate(X)  ;
\fill[orange](X) circle (3.5pt);
 \draw[cyan,line width=0.08cm](01.5,0)--++(90:1.5)  ;
 
  \draw[magenta,line width=0.08cm](2,0)--++(90:0.4) coordinate(X)  ;
\fill[magenta](X) circle (3.5pt);
 
 
 
 
  \draw[darkgreen,line width=0.08cm](2.5,0)--++(90:0.55) ;
  \draw[darkgreen,line width=0.08cm] (0.5,1.05) --++(0:0.5) to [out=0,in=90] (2.5,0.55) ;
 
\draw(2.5,0) node [below] {$ \bf  \color{darkgreen}\sf D^1$};
 
 
 \end{tikzpicture}   $$ 


 
  \begin{defn} (B.-D.-H.-N. \cite{compan}, Libedinsky--Williamson \cite{MR4437613})
  \label{basis}
  Let $(W,P)=(\mathfrak{S}_{n+m},\mathfrak{S}_{n}\times \mathfrak{S}_{m})$. 
The algebra $\mathscr{H}_{(W,P)}$  has a graded cellular basis given by 
  $\{c_\SSTS^\ast c_\SSTT\, : \, \SSTS, \SSTT \in {\rm Path}(\la), \, \la \in \mptn\}$ with ${\rm deg}(c_\SSTS^\ast c_\SSTT) = {\rm deg}(\SSTS) + {\rm deg}(\SSTT)$ with respect to the poset $(\mptn , \leq)$ and the anti-involution $\ast$. The multiplication is given by vertical concatenation subject to the following local relations together with their horizontal and vertical flips:
$$\begin{minipage}{1cm}\begin{tikzpicture}[scale=1]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
\draw[magenta,line width=0.08cm](0,0)--++(90:1.5);
\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
\draw[magenta,line width=0.08cm](0,0.3) to [out=0,in=180](0.5,1.2);
  \end{tikzpicture}
  \end{minipage}= 
  \begin{minipage}{1cm}\begin{tikzpicture}[scale=1]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
\draw[magenta,line width=0.08cm](0.25,0.4)--++(90:1.5-0.8);
%\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
\draw[magenta,line width=0.08cm](0,0) to [out=90,in=180]
(0.25,0.4)  to [out=0,in=90]
(0.5,0);

\draw[magenta,line width=0.08cm](0,1.5) to [out=-90,in=180]
(0.25,1.5-0.4)  to [out=0,in=-90]
(0.5,1.5);
  \end{tikzpicture}
  \end{minipage}
  \qquad
  \quad   \quad 
  \begin{minipage}{1 cm}\begin{tikzpicture}[scale=1 ]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
\draw[magenta,line width=0.08cm](0,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
\draw[magenta,line width=0.08cm](0,0.3) to [out=0,in=-90](0.5,0.75)--++(90:0.4) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
  \end{tikzpicture}
  \end{minipage}= 
    \begin{minipage}{1. cm}\begin{tikzpicture}[scale=1 ]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
\draw[magenta,line width=0.08cm](0,0)--++(90:1.5);
%%\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0,0.3) to [out=0,in=-90](0.5,0.75)--++(90:0.4) coordinate(hi);
%\fill[magenta] (hi) circle (3.5pt);
  \end{tikzpicture}  \end{minipage}
  \qquad  \quad 
  \quad
    \begin{minipage}{1 cm}\begin{tikzpicture}[scale=1 ]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
\draw[magenta,line width=0.08cm](0,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
\draw[magenta,line width=0.08cm](0,0.3) to [out=0,in=-90](0.5,0.75); 
\draw[magenta,line width=0.08cm](0,1.5-0.3) to [out=0,in=90](0.5,0.75);

   \end{tikzpicture}
  \end{minipage}
  =\;
  0 \qquad\quad
%\end{equation}
%together with the ``one colour Demazure" relation 
%\begin{equation}\tag{S2}\label{S2}
\begin{minipage}{1cm}\begin{tikzpicture}[scale=1]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
\draw[magenta,line width=0.08cm](0.25,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0,0.3) to [out=0,in=180](0.5,1.2);
\draw[magenta,line width=0.08cm] (0.5,0.75)--++(90:0.3) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
\draw[magenta,line width=0.08cm] (0.5,0.75)--++(-90:0.3) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
  \end{tikzpicture}
  \end{minipage}
  =\; 
  2 \; 
  \begin{minipage}{1cm}\begin{tikzpicture}[scale=1]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
%\draw[magenta,line width=0.08cm](0.25,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0,0.3) to [out=0,in=180](0.5,1.2);
\draw[magenta,line width=0.08cm] (0.25,0)--++(90:0.4) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
\draw[magenta,line width=0.08cm] (0.25,1.5)--++(-90:0.4) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
  \end{tikzpicture}
  \end{minipage}
  -\; 
  \begin{minipage}{1cm}\begin{tikzpicture}[scale=1]
\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,1.5) ;  
\draw[magenta,line width=0.08cm](0.25,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0.5,0)--++(90:1.5);
%\draw[magenta,line width=0.08cm](0,0.3) to [out=0,in=180](0.5,1.2);
\draw[magenta,line width=0.08cm] (0 ,0.75)--++(90:0.3) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
\draw[magenta,line width=0.08cm] (0 ,0.75)--++(-90:0.3) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
  \end{tikzpicture}
  \end{minipage} 
$$  and  
for $m_{\csigma \ctau}=3 $ we have the {\em  2-colour barbell  relation},
\begin{align*} 
 \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]
\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
\draw[line width=0.08cm, magenta, fill=cyan](0,0)--++(90:3) 
  circle (4pt);
  \path(0.55,0) coordinate (hi) ;
 \draw[line width=0.08cm, cyan, fill=cyan](hi)--++(90:0.4) 
  circle (4pt);
  \draw[line width=0.08cm, cyan, fill=cyan](hi)--++(-90:0.4) 
  circle (4pt);
   \draw[line width=0.08cm, magenta, fill=cyan](0,0)--++(-90:3) 
  circle (4pt);
\end{tikzpicture}
\end{minipage}
\;=\;
\begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]
\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
\draw[line width=0.08cm, magenta, fill=cyan](0,0)--++(90:3) 
  circle (4pt);
\path(-0.55,0) coordinate (hi) ;
 \draw[line width=0.08cm, cyan, fill=cyan](hi)--++(90:0.4) 
  circle (4pt);
  \draw[line width=0.08cm, cyan, fill=cyan](hi)--++(-90:0.4) 
  circle (4pt);
  \draw[line width=0.08cm, magenta, fill=cyan](0,0)--++(-90:3) 
  circle (4pt);
\end{tikzpicture} 
\end{minipage}\;- 
 \begin{minipage}{1.3cm}\begin{tikzpicture}[xscale=-0.65,yscale=0.65]
\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
\draw[line width=0.08cm, magenta, fill=magenta](0,0)--++(90:3) 
  circle (4pt);
  \path(0.55,0) coordinate (hi) ;
 \draw[line width=0.08cm, magenta, fill=magenta](hi)--++(90:0.4) 
  circle (4pt);
  \draw[line width=0.08cm, magenta, fill=magenta](hi)--++(-90:0.4) 
  circle (4pt);
   \draw[line width=0.08cm, magenta, fill=magenta](0,0)--++(-90:3) 
  circle (4pt);
\end{tikzpicture}
\end{minipage}
\;+   \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]
\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
\draw[line width=0.08cm, magenta, fill=magenta]
(0,3)--(0,0.5) 
  circle (4pt);
  \draw[line width=0.08cm, magenta, fill=magenta]
  (0,-3)--(0,-0.5) 
  circle (4pt);
 \end{tikzpicture}
\end{minipage}
\end{align*} 
  and for   $m_{\csigma \ctau}=3$ and  $m_{\ctau \crho}=2 $ we have  the {\em  Temperley--Lieb relations}, 
$$
\begin{minipage}{1.5cm}
 \begin{tikzpicture}[scale=0.9]

\draw[densely dotted, rounded corners] (-0.25,0) rectangle (1.25,2);

\begin{scope}

\clip(-0.25,0) rectangle (1.25,2); 
  

 \draw[magenta ,line width=0.08cm](0,0)    -- (0,2);
 \draw[magenta ,line width=0.08cm](1,0)    -- (1,2); 

 \draw[cyan ,line width=0.08cm](0.5,0)    -- (0.5,2);
 

  \end{scope}
 

 \end{tikzpicture}
 \end{minipage}
 = - \;
    \begin{minipage}{1.5cm}
 \begin{tikzpicture}[scale=0.9 ]

\draw[densely dotted, rounded corners] (-0.25,0) rectangle (1.25,2);

\begin{scope}

\clip(-0.25,0) rectangle (1.25,2); 
%   \draw[magenta ,line width=0.08cm](0,0)    -- (0,2);
%
% \draw[magenta ,line width=0.08cm](1,0)   to [out=90,in=-15]   (0,0.8);
% \draw[magenta ,line width=0.08cm](1,2)   to [out=-90,in=15]   (0,2-0.8);

 \draw[magenta ,line width=0.08cm](1,0)   to [out=90,in=0] (0.5,0.7) to [out=180,in=90]  (0,0);
 
 
  \draw[magenta ,line width=0.08cm](1,2)   to [out=-90,in=0] (0.5,2-0.7) to [out=180,in=-90]  (0,2);

  \draw[magenta ,line width=0.08cm](0.5,0.7)    -- (0.5,2-0.7);
 
 

\draw[cyan ,line width=0.08cm](0.5,0)   to (0.5,0.4) coordinate (hi); 
\fill[cyan ,line width=0.08cm] (hi) circle (3pt); 


\draw[cyan ,line width=0.08cm](0.5,2)   to (0.5,2-0.4) coordinate (hi); 
\fill[cyan ,line width=0.08cm] (hi) circle (3pt); 
 
  \end{scope}
 

 \end{tikzpicture} \end{minipage} 
 \qquad\quad\quad\quad
 \begin{minipage}{1cm}
 \begin{tikzpicture}[scale=0.9]

\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,2);

\begin{scope}

\clip(-0.25,0) rectangle (0.75,2); 
  
 
 \draw[darkgreen ,line width=0.08cm](0,0)  to  	 (0,2);
 
 \draw[cyan ,line width=0.08cm](0.5,0)   to  (0.5,2);
 


  \end{scope}
 

 \end{tikzpicture}
 \end{minipage}
 \  = \;   \begin{minipage}{1cm}
 \begin{tikzpicture}[scale=0.9]

\draw[densely dotted, rounded corners] (-0.25,0) rectangle (0.75,2);

\begin{scope}

\clip(-0.25,0) rectangle (0.75,2); 
  

 \draw[darkgreen ,line width=0.08cm](0,0)  to [out=90,in=-90]
 (0.5,1)  		to [out=90,in=-90]			 (0,2);
 
 \draw[cyan ,line width=0.08cm](0.5,0)   to [out=90,in=-90] (0,1) 
 to [out=90,in=-90] (0.5,2);
  
  \end{scope}
 

 \end{tikzpicture}\end{minipage} 
$$
   and for 
  $m_{\ctau \crho}=m_{ \ctau \cpi}=m_{\crho \cpi}=2 $  
 the {\em  commutativity relations,} 
$$
   \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]
\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
 \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(-45:3.5) 
  circle (4pt);  \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(135:3.5) 
  circle (4pt);
   \draw[line width=0.08cm, darkgreen, fill=darkgreen](0,0)--++(-135:0.7) 
   circle (4pt);
  
  
     
    \draw[line width=0.08cm, darkgreen, fill=darkgreen](0,0)--++(45:3.5) coordinate(hi)
 ;
 

\end{tikzpicture}\end{minipage} \; =
\;   
   \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]
\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
 \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(-45:3.5) 
  circle (4pt);  \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(135:3.5) 
  circle (4pt);
   \path (0,0)--++(45:30pt) 
   coordinate (hi);
  
  
     
    \draw[line width=0.08cm, darkgreen, fill=darkgreen](hi)--++(-135:18pt)  circle (4pt)
 ;
 

\end{tikzpicture}\end{minipage}
\qquad\quad
   \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
 \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(-60:3.5) 
  circle (4pt); 
     
  
  
     \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(120:3.5) 
  circle (4pt);
    \draw[line width=0.08cm, darkgreen, fill=darkgreen](0,0)--++(30:0.6) coordinate(hi)
 ;
 \draw[line width=0.08cm, darkgreen, fill=darkgreen](0,0)--++(-150:3.5) 
  circle (4pt);

    \draw[line width=0.08cm, darkgreen, fill=darkgreen] (hi)--++(90:1);
        \draw[line width=0.08cm, darkgreen, fill=darkgreen] (hi)--++(-30:1);
 ;

\end{tikzpicture}
\end{minipage}\; =\; 
   \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]
   \draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
      
    
    \path[line width=0.08cm, darkgreen, fill=darkgreen](0,0)--++(30:0.6) coordinate(hi)
 ;
 \draw[line width=0.08cm, darkgreen, fill=darkgreen](0,0)--++(-150:3.5) 
  circle (4pt);

    \path (hi)--++(90:17.5pt) coordinate (hi2);
        \path (hi)--++(-30:17.5pt) coordinate (hi3);
 ;
 
    \path (0,0)--++(-150:7pt) coordinate (hi4);
 ;
 
\draw[line width=0.08cm, darkgreen, fill=white]  (hi3) -- (hi4)--(hi2); 

\path(0,0)--++(120:30pt) coordinate (hiya);
\path(0,0)--++(-60:30pt) coordinate (hiya2);
 \draw[line width=0.08cm, cyan](hiya) to [out=-30,in=90] (hiya2); 



\end{tikzpicture}
\end{minipage}
 \qquad\quad
   \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
 \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(-45:3.5) 
  circle (4pt);  \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(135:3.5) 
  circle (4pt);
   \draw[line width=0.08cm, darkgreen, fill=orange](0,0)--++(-135:3.5) 
   circle (4pt);
     \draw[line width=0.08cm, darkgreen](0,0)--++(45:3.5) coordinate(hi)
 ;
  \draw[line width=0.08cm, orange] (32pt,0) to [out=-135, in=-45] (-32pt,0);  
\end{tikzpicture}\end{minipage}=
   \begin{minipage}{1.3cm}\begin{tikzpicture}[scale=0.65]\draw[densely dotted](0,0)  circle (30pt);
\clip(0,0) circle (30pt);
 \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(-45:3.5) 
  circle (4pt);  \draw[line width=0.08cm, cyan, fill=cyan](0,0)--++(135:3.5) 
  circle (4pt);
   \draw[line width=0.08cm, darkgreen, fill=orange](0,0)--++(-135:3.5) 
   circle (4pt);
      \draw[line width=0.08cm, darkgreen](0,0)--++(45:3.5) coordinate(hi)
 ;
    \draw[line width=0.08cm, orange] (32pt,0) to [out=135, in=45] (-32pt,0);  
  \end{tikzpicture}\end{minipage}
 $$
%  
%  
% 
% \begin{rmk}
%We emphasise that  any of these ``local" relations can be applied to an arbitrary 
%region of the diagram. 
% 
% \end{rmk}
%
%
%  
Finally, we have the non-local  {\em  cyclotomic relations,}   
$$\begin{minipage}{0.7cm}\begin{tikzpicture}[scale=1.2]
\draw[densely dotted, rounded corners] (0.25,0.25) rectangle (0.75,1.25) ;  
  \draw[cyan,line width=0.08cm] (0.5,0.75)--++(90:0.5) coordinate(hi);
%\fill[cyan] (hi) circle (3.5pt);
\draw[cyan,line width=0.08cm] (0.5,0.75)--++(-90:0.5) coordinate(hi);
%\fill[cyan] (hi) circle (3.5pt);
  \end{tikzpicture}
  \end{minipage}
  \otimes {\sf1}_\w =0 
\qquad
\quad
 \begin{minipage}{0.7cm}\begin{tikzpicture}[scale=1.2]
\draw[densely dotted, rounded corners] (0.25,0.25) rectangle (0.75,1.25) ;  
  \draw[magenta,line width=0.08cm] (0.5,0.75)--++(90:0.3) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
\draw[magenta,line width=0.08cm] (0.5,0.75)--++(-90:0.3) coordinate(hi);
\fill[magenta] (hi) circle (3.5pt);
  \end{tikzpicture}
  \end{minipage}
  \otimes {\sf1}_\w =0 
$$
for $\csigma \in S_W$,  $\ctau \in S_P$,
 and $\w$ an arbitrary  word for some $w\in W$.  
\end{defn}











The following theorems will hold true in the setting of arbitrary parabolic Coxeter systems $(W,P)$.  Thus we state them in that language (lifting the combinatorics from \cref{reduced-defn}) despite the fact that we have only provided  the (much simplified!) relations of the case $(W,P)=(\mathfrak{S}_{n+m},\mathfrak{S}_{n}\times \mathfrak{S}_{m})$.

For $\la \in {^PW}$ we let $ \mathscr{H}_{(W,P)}^{<\la} $ denote the span of all diagrams $c_\SSTS^\ast c_\SSTT$ with $\SSTS , \SSTT \in {\rm Path}(\mu)$ with $\mu <\la$.



 \begin{thm}[The light leaves basis \cite{MR4437613}]
For each  $\la\in {^PW}$ the  graded cell module $\Delta^\Bbbk(\la)$ has a basis given by 
 $$\{ c_\SSTS + \mathscr{H}_{(W,P)}^{<\la} \mid \SSTS \in \Path(\la)\}$$
 This module has a unique proper maximal submodule, ${\rm rad}(\Delta^\Bbbk(\la))$, with simple quotient
 $$L^\Bbbk(\la)=\Delta^\Bbbk(\la)/ {\rm rad}(\Delta^\Bbbk(\la))$$
Moreover, the set $\{ L^\Bbbk(\la)\langle k \rangle \mid \la\in {^PW}, \, k\in \mathbb{Z} \}$ provides a complete set of pairwise non-isomorphic graded simple modules for $\mathscr{H}_{(W,P)}$.
 \end{thm}
 
 
 \begin{thm}[The Kazhdan--Lusztig positivity conjecture, Elias--Williamson 
 \cite{MR3245013}]
  Let $\Bbbk$ be a field of characteristic $p\geq 0$. 
 The $p$-Kazhdan--Lusztig polynomials are defined to be the graded composition factor multiplicities 
 $${^p}n_{\la,\mu}(q) = \sum _{k \in \ZZ}[\Delta^\Bbbk(\la): L^\Bbbk(\mu)\langle k \rangle] q^k.$$
 For $p=0$ we have that the ${^p}n_{\la,\mu}(q)$ specialise to the classical Kazhdan--Lusztig polynomials of \cref{reduced-defn} and thus the  classical Kazhdan--Lusztig polynomials have non-negative coefficients. 
 \end{thm}
 
 
 
 
 
 
 
 
 
  \section{Partitions and their Dyck combinatorics }








Formally, a {\em partition}    $\lambda $ of $\ell$  is defined to be a weakly decreasing  sequence   of non-negative integers $\lambda = (\lambda_1, \lambda_2, \ldots )$ which  sum to $\ell$.  We call $\ell (\lambda) := \ell = \sum_{i}\lambda_i$ the length of the partition $\lambda$.
We define the Young diagram of a partition to be the collection of tiles 
$$[\la]=\{[r,c] \mid 1\leq c \leq \la_r\}$$
depicted in Russian style with rows at $135^\circ$ and columns at $45^\circ$ (as in \cref{fig3}).  We identify a partition with its Young diagram and we write $\la\subseteq \mu $ if every box of $\la$ is contained in $\mu$ (that is $\la_i\leq \mu_i$ for all $i\geq1$).
We let $\la^t$ denote the transpose partition given by reflection 
of the Russian Young diagram through the vertical axis.  
 Given  $m,n\in \NN$ we let  ${\mathscr P}_{m,n}$ denote the set of all partitions which fit into an $m\times n$ rectangle, that is 
$${\mathscr P}_{m,n}= \{ \la \mid \la_1\leq m, \la_1^t \leq n\}.$$
For $\lambda \in {\mathscr P}_{m,n}$, the $x$-coordinate of a tile $[r,c] \in \lambda$ is   equal to   $r-c+m \in \{1,2,\dots, m+n\}$ and we define this $x$-coordinate to be  the  ``colour" or ``content" of the tile and we  write ${\mathsf cont}[r,c]=r-c+m$.  It is well-known that a partition is uniquely determined by the contents of its boxes and this can be seen as the main ingredient in the following result:


\begin{prop}
For $(W,P)=(\mathfrak{S}_{n+m},\mathfrak{S}_{n}\times \mathfrak{S}_{m})$ there is a poset  isomorphism between $({^PW}, \leq)$ (the minimal coset representatives under the Bruhat ordering) and $({\mathscr P}_{m,n}, \leq )$ (the partitions in an $(m\times n)$-rectangle ordered by inclusion), sending the identity coset to $\varnothing$ and the longest element to $(m^n)$ (see \cref{Bruhatexample-KL,fig3}). 
\end{prop}




 
\begin{figure}[ht!]
$$\begin{tikzpicture} [yscale=0.4,xscale=-0.4]

 
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\draw[very thick, fill=orange] ( -10,3-2.5) coordinate circle (7pt); 

\draw[very thick, fill=darkgreen] ( -8,3-2.5) coordinate circle (7pt);  

\draw[very thick, fill=magenta] ( -6,3-2.5) coordinate circle (7pt); 
\draw( -7,3-1.25) node {$\mathfrak{S}_2 \times \mathfrak{S}_3 \leq \mathfrak{S}_5 $};
 
\draw[very thick, fill=cyan] ( -4,3-2.5) coordinate circle (7pt); 

\draw[very thick, rounded corners] (-10.8,2.5-2.5) rectangle (-7.2,3.5-2.5); 

\draw[very thick, rounded corners] (-4.8,3.5-2.5) rectangle (-3.2,2.5-2.5); 










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\path[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);
\path[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);
\path[magenta, line width=3] (hi4)--(6,10.5) coordinate (hi7);


\path[magenta, line width=3] (hi6)--(2,13) coordinate (hi8);
\path[orange, line width=3] (hi7)--(2,13) coordinate (hi8);
\path[darkgreen, line width=3] (hi8)--(2,15) ; 


 \path (2,-1) coordinate (top); 
 
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\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);









  
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 \path (2,-1) coordinate (top); 
 
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\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
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   \draw[thick] (X1)--++(135:0.4*3);
  \draw[thick] (Y2)--++(45:0.4*2);
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 \draw [  thick,fill=magenta!80] (top)--++(45:0.4*1)--++(135:0.4)--++(-135:0.4)--++(-45:0.4);
 
%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);

















  
\draw[darkgreen, line width=3] (hi)--(-2,5.5) coordinate (hi3);
 
\path (2,2.5)  coordinate (top); 
 
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\draw[thick, fill=white](circle) circle (28pt);
\path(top) --++(45:0.5*0.4) coordinate(top);



\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
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--++(-45:1*0.4) --++(45:0.4)    ;  
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  \draw[thick] (Y1)--++(45:0.4*2);    
 \draw [  thick,fill=magenta!80] (top)--++(45:0.4*1)--++(135:0.4)--++(-135:0.4)--++(-45:0.4);
 
%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);

 \path (-2,5)  coordinate (top); 
 
\path(top)--++(45:0.4*1.5)--++(135:0.4*1.5)  coordinate (circle);
\draw[thick, fill=white](circle) circle (28pt);
\path(top) --++(45:0.5*0.4) coordinate(top);



\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
--++(135:3*0.4)--++(-135:2*0.4)--++(-45:1*0.4)  coordinate (Y2) 
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 \draw [  thick,fill=darkgreen] (TL)--++(135:0.4*1) coordinate (TL)--++(135:0.4)--++(-135:0.4)--++(-45:0.4)--++(45:0.4);
 
%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);










  
\draw[cyan, line width=3] (hi)--(6,5.5) coordinate (hi2);

 

\path (2,2.5)  coordinate (top); 
 
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\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
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% --++(-45:0.4*2);

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%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);




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%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);


































  

\draw[cyan, line width=3] (hi5)--(-2,10.5) coordinate (hi6);
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% --++(-45:0.4*2);

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% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);




 \path (-2,10)  coordinate (top); 
 
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\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
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%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);





  

%\draw[orange, line width=3] (hi4)--(-2,10.5) coordinate (hi6);
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% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);








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% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);








 








 \path (6,10)  coordinate (top); 
 
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\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
--++(135:3*0.4)--++(-135:2*0.4)--++(-45:1*0.4)  coordinate (Y2) 
--++(-45:1*0.4)  coordinate (Y1) 
--++(-45:1*0.4) --++(45:0.4)    ;  
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 \draw [  thick,fill=darkgreen!80] (TR)--++(45:0.4*1)  --++(135:0.4)
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%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);

 




















  


 
\draw[magenta, line width=3] (hi6)--(2,13) coordinate (hi8);
\draw[orange, line width=3] (hi7)--(2,13) coordinate (hi8);
%\draw[darkgreen, line width=3] (hi8)--(2,15) ; 



 \path (-2,10)  coordinate (top); 
 
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\draw[thick, fill=white](circle) circle (28pt);
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% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);


 \path (6,10)  coordinate (top); 
 
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\draw[thick, fill=white](circle) circle (28pt);
\path(top) --++(45:0.5*0.4) coordinate(top);

 
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% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);

 



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\path(top) --++(45:0.5*0.4) coordinate(top);

 
\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
--++(135:3*0.4)--++(-135:2*0.4)--++(-45:1*0.4)  coordinate (Y2) 
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--++(-45:1*0.4) --++(45:0.4)    ;  
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 \draw [  thick,fill=darkgreen!80] (TR)--++(45:0.4*1)  --++(135:0.4)
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  \draw [  thick,fill=orange!80] (TR)--++(45:0.4*1)  --++(135:0.4)--++(-135:0.4) --++(-45:0.4);


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%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);




  


\draw[darkgreen, line width=3] (hi8)--(2,16) ; 

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 \draw [  thick,fill=darkgreen!80] (TR)--++(45:0.4*1)  --++(135:0.4)
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%\draw [  thick,fill=gray!40] (top)--++(45:0.4*2)--++(135:0.4)--++(-135:0.4)--++(135:0.4)--++(-135:0.4)
% --++(-45:0.4*2);

\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);







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\draw[thick, fill=white](circle) circle (28pt);
\path(top) --++(45:0.5*0.4) coordinate(top);

 
\draw[ thick]  (top)--++(45:1*0.4) coordinate (X1)--++(45:1*0.4) coordinate (X2) 
--++(135:3*0.4)--++(-135:2*0.4)--++(-45:1*0.4)  coordinate (Y2) 
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--++(-45:1*0.4) --++(45:0.4)    ;  
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 \draw [  thick,fill=magenta!80] (top)--++(45:0.4*1) coordinate (TL) --++(135:0.4)--++(-135:0.4) coordinate (TR)--++(-45:0.4);

 \draw [  thick,fill=darkgreen!80] (TR)--++(45:0.4*1)  --++(135:0.4)
 --++(-135:0.4)  coordinate (TR)--++(-45:0.4);
 

  \draw [  thick,fill=orange!80] (TR)--++(45:0.4*1) coordinate (X)  --++(135:0.4) --++(-135:0.4) --++(-45:0.4);

  \draw [  thick,fill=darkgreen!80]   (X)--++(45:0.4*1)  --++(135:0.4)--++(-135:0.4) --++(-45:0.4);



  \draw [  thick,fill=cyan!80] (TL)--++(45:0.4*1)  --++(135:0.4)--++(-135:0.4)coordinate (TR)--++(-45:0.4);
 

  \draw [  thick,fill=magenta!80] (TR)--++(45:0.4*1) coordinate (TL)--++(135:0.4)--++(-135:0.4)--++(-45:0.4);

 
 
\draw [  thick] (top)--++(45:0.4)--++(135:0.4)--++(-135:0.4);


 
 
\end{tikzpicture}
$$  
\caption{The partitions in a $(2\times 3)$-rectangle, ordered by inclusion.  
At the bottom we depict the empty partition inside a $(2\times 3)$-grid and at the top we depict the unique partition of maximal size, namely the rectangle $(2^3)$.  
Compare this poset with the rightmost poset depicted in \cref{Bruhatexample-KL}.    }
\label{fig3}
\end{figure}



Having encoded the Bruhat order in terms of partition combinatorics, we ask whether it is possible compute the Kazhdan--Lusztig polynomials in a similar fashion.
The answer is yes, and makes use of the idea of Dyck paths.  
 We define a path on $\la$ to    
be a finite non-empty set $P$
 of  tiles  that are ordered $[r_1, c_1] \in \la, \ldots , [r_s, c_s] \in \la$ for some $s\geq 1$ such that 
 for each $1\leq i\leq s-1$ we have $[r_{i+1}, c_{i+1}] = [r_i+1, c_i]$ or $[r_i, c_i -1]$. Note that the set ${\sf cont}(P)$ of contents of the tiles in a path $P$ form an interval of integers.
 We say that $P$ is a {\sf Dyck path} if
 $$\min \{ r_i+c_i \, : \, 1\leq i\leq s\} = r_1+c_1 = r_s+c_s,$$
that is the minimal height of the path is achieved at the start and end of the path, see the leftmost diagram in \cref{Dyck!} for an example of a single Dyck path on a partition.  
We say that $P$ and $Q$ are {\em adjacent} if and only if the multiset given by the disjoint union ${\sf cont}(P) \sqcup {\sf cont}(Q)$ is an interval (see the central diagram in \cref{Dyck!} for an example).  

 \begin{defn}\label{Dyckpair}
 Let $\lambda\subseteq  \mu\in \mptn$. A {\sf Dyck tiling} of the skew partition $\mu\setminus \la$ is a set $\{P^1,\dotsc,P^k\}$ of Dyck paths such that
 $$\mu\setminus \la = \bigsqcup_{i=1}^k P^i $$
 and for each $i\neq j$ we have   $P^i$ and  $P^j$ are not adjacent.  
 If such a Dyck tiling exists, we call $(\la,\mu)$ a {\sf Dyck pair}.  Dyck tilings for a given $\mu \setminus \la$ are not unique. However, it can be shown that if we have two Dyck tilings $\mu \setminus \la = \sqcup_{i=1}^k P^i = \sqcup_{j=1}^{l}Q^j$ then we must have $k=l$ and there is a bijection $\{P^i\} \rightarrow \{Q^{j_i}\}$ satisfying ${\sf cont}(P^i) = {\sf cont}(Q^{j_i})$ for all $1\leq i \leq k$. Thus it makes sense to define the {\sf degree} of  the Dyck pair $(\la, \mu)$ to be  ${\sf deg}(\la, \mu) = k$.
 \end{defn}














  



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\caption{On the left we depict a  Dyck path on $    (9^6,6^3 )$.
The centre diagram depicts two adjacent Dyck paths (and so $ (9^6,6^3 ) \setminus (9^2,8^3,5^3,3 )$ does not admit a Dyck tiling). 
On the right we depict a Dyck tiling of 
$
    (9^6,6^3 )
    \setminus (9,7,6,5,4,2,1^2 )$ of degree 6.}
    \label{Dyck!}
\end{figure}

































We are now ready to provide a closed combinatorial interpretation for the $p$-Kazhdan--Lusztig polynomials of $(\mathfrak{S}_{n+m},\mathfrak{S}_{n}\times \mathfrak{S}_{m})$.  This  generalises existing results of Lascoux--Schutzenberger to arbitrary fields. 

\begin{thm}
[B.--D.--H.--Norton \cite{compan}]
\label{char-free}
Let $(W,P)=(\mathfrak{S}_{n+m},\mathfrak{S}_{n}\times \mathfrak{S}_{m})$ and $p\geq 0$. We have 
that $${^p}n_{\la,\mu}(q)
=
\begin{cases}
q^{\deg(\la , \mu)} &\text{if $(\la, \mu)$ is a Dyck pair;} \\
0	&\text{otherwise.}
\end{cases}$$
\end{thm}



\begin{proof}
% Thus, %to show that the $p$-Kazhdan--Lusztig polynomials are characteristic-free, 
%if we can show that $\mathscr{H}_{(W,P)}$ is generated by idempotents (of degree 0)
% and elements of degree 1 then we can conclude that all the simple modules are 1-dimensional (concentrated in degree zero).  
By \cref{basis} we know that $\mathscr{H}_{(W,P)}$  has basis indexed by pairs of paths in the weak 
Bruhat graph of $^PW$.  
 In \cite{compan} we provide a graded bijection between  $\Path(\la,\SSTT^\mu)$  
  and  Dyck tilings of shape $\mu \setminus \la$.
Any Dyck tiling $\mu \setminus \la$ is manifestly of {\em positive degree}, unless $\la=\mu$ in which case we obtain a unique (trivial) Dyck tableau of degree zero.  
Now, since any graded simple module is fixed by the anti-involution $\ast$ we deduce that it must have graded dimension belonging to $\ZZ_{\geq 0}[q+q^{-1}]$.
 Putting together the above facts, we deduce that the 
  simple modules are 1-dimensional (concentrated in degree zero) regardless of the characteristic of the field and the result follows.
%   We hence deduce that % the basis of  $\mathscr{H}_{(W,P)}$ is non-negatively graded 
%%  and that
% the unique   element of non-positive degree in $\Path(\la,\SSTT^\mu)$ is the trivial 
%% 
%%  and we prove that the corresponding basis element can be written as a product of 
 \end{proof}





\section{Submodule lattices of cell modules}

We are now ready to discuss one of the main results of \cite{compan2}.  Namely, we will provide the full submodule lattice of the cell modules  for $\mathscr{H}_{(W,P)}$ when $(W,P)=(\mathfrak{S}_{n+m},\mathfrak{S}_{n}\times \mathfrak{S}_{m})$ over any field $\Bbbk$.
We prove in 
\cite{compan2}
 that  (the  basic algebra of) $\mathscr{H}_{(W,P)}$  is   generated in degrees 0 and 1 and hence the grading gives a submodule filtration of 
 $\Delta^\Bbbk(\la)$.    
 Thus to determine whether there is an extension between two composition factors $L^\Bbbk(\mu)$ and $L^\Bbbk(\nu)$ within  $\Delta^\Bbbk(\la)$ (where $(\la, \mu)$ and $(\la ,\nu)$ are Dyck pairs, by \cref{char-free}) it is enough to consider pairs of adjacent degree , that is where $\deg( \la , \nu)=\deg( \la , \mu ) +1$.  
Using the presentations of \cite[Theorem B]{compan2} we are able to fully determine these extensions combinatorially as follows:









\begin{defn} Let $(\la, \mu)$ and $(\la, \nu)$ be Dyck pairs of degree $k$ and $k+1$ respectively. 
 We write 
$(\la, \mu) \to (\la, \nu)$ if either:
\begin{itemize}
  \item $\nu$ is obtained from $\mu$ by adding a Dyck path.  
$$
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\fill[brown] (start) circle (2pt);
 


\draw [very thick] (L1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);


\path(L1)--++(-90:0.21)node  {};
\draw [very thick] (L1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);



\end{tikzpicture}\end{minipage}
$$

 
\item $\nu$ is obtained from $\mu$ by removing a Dyck path,  splitting some Dyck  path in the tiling of $\mu \setminus \la$ into two distinct Dyck paths: 
$$
\begin{minipage}{2.25cm}
\begin{tikzpicture}[yscale=1.001,xscale=-1.001]
 


\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
--++(45:1*0.3)  coordinate (X3) 
--++(45:1*0.3) 
--++(135:4*0.3)--++(-135:4*0.3)--++(-45:1*0.3)  coordinate (Y3) 
--++(-45:1*0.3)  coordinate (Y2) 
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;  
 \draw[densely dotted] (X1)--++(135:0.3*4);
  \draw[densely dotted] (X2)--++(135:0.3*4);
    \draw[densely dotted] (X3)--++(135:0.3*4);
      \draw[densely dotted] (Y1)--++(45:0.3*4);
    \draw[densely dotted] (Y2)--++(45:0.3*4);
        \draw[densely dotted] (Y3)--++(45:0.3*4);
        
        
         
    
\draw [very thick,fill=white] (L1)--++(45:0.3*4)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)
 --++(-135:0.3)--++(135:0.3)
--++(-135:0.3*2)
 --++(-45:0.3*3);

    \draw [very thick,fill=gray!40] 
          (L1)--++(45:0.3*3)--++(135:0.3*1)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*1)--++(135:0.3)--++(-135:0.3*1)
 --++(-45:0.3*3);
             
             
             
             
               \draw[very thick] (Y3)--++(45:0.3*2);               \draw[very thick] (Y2)--++(45:0.3*3);
               \draw[very thick] (Y1)--++(45:0.3*4);               
         \draw[very thick] (X3)--++(135:0.3*2);               \draw[very thick] (X2)--++(135:0.3*3);
               \draw[very thick] (X1)--++(135:0.3*4);               



\path(L1)--++(45:0.5*0.3)--++(135:3.5*0.3) coordinate (start);

\fill[cyan] (start) circle (1.8pt);
\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);

\fill[cyan] (start) circle (1.8pt);
\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[cyan] (start) circle (2pt);
 
\fill[cyan] (start) circle (1.8pt);
\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);

\fill[cyan] (start) circle (1.8pt);
\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[cyan] (start) circle (2pt);


\fill[cyan] (start) circle (1.8pt);
\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);

\fill[cyan] (start) circle (1.8pt);
\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[cyan] (start) circle (2pt);



\draw [very thick] (L1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (L1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);

  \path(L1)--++(-90:0.21) node {};


\end{tikzpicture}
\end{minipage}
\xrightarrow {\ \ \ \ \ }
\;\;
\begin{minipage}{2.25cm}
\begin{tikzpicture}[yscale=1.001,xscale=-1.001]
 


\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
--++(45:1*0.3)  coordinate (X3) 
--++(45:1*0.3) 
--++(135:4*0.3)--++(-135:4*0.3)--++(-45:1*0.3)  coordinate (Y3) 
--++(-45:1*0.3)  coordinate (Y2) 
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;  
 \draw[densely dotted] (X1)--++(135:0.3*4);
  \draw[densely dotted] (X2)--++(135:0.3*4);
    \draw[densely dotted] (X3)--++(135:0.3*4);
      \draw[densely dotted] (Y1)--++(45:0.3*4);
    \draw[densely dotted] (Y2)--++(45:0.3*4);
        \draw[densely dotted] (Y3)--++(45:0.3*4);
        
        
         
    
\draw [very thick,fill=white] (L1)--++(45:0.3*4)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)
 --++(-135:0.3)--++(135:0.3)
--++(-135:0.3*2)
 --++(-45:0.3*3);

    \draw [very thick,fill=gray!40] 
          (L1)--++(45:0.3*3)--++(135:0.3*1)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*1)--++(135:0.3)--++(-135:0.3*1)
 --++(-45:0.3*3);
             
             
             
             
               \draw[very thick] (Y3)--++(45:0.3*2);               \draw[very thick] (Y2)--++(45:0.3*3);
               \draw[very thick] (Y1)--++(45:0.3*4);               
         \draw[very thick] (X3)--++(135:0.3*2);               \draw[very thick] (X2)--++(135:0.3*3);
               \draw[very thick] (X1)--++(135:0.3*4);               



\path(L1)--++(45:0.5*0.3)--++(135:3.5*0.3) coordinate (start);

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(45:0.3) coordinate (start);

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[magenta] (start) circle (2pt);
 
\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(45:0.3) coordinate (start);

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[magenta] (start) circle (2pt);

%
%\fill[cyan] (start) circle (1.8pt);
 \path (start)--++(45:0.3) coordinate (start);

%\fill[cyan] (start) circle (1.8pt);
\path(start)--++(-45:0.3) coordinate (start);

\fill[orange] (start) circle (2pt);



\draw [very thick] (L1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (L1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);

  \path(L1)--++(-90:0.21) node {};


\end{tikzpicture}
\end{minipage}
\;\;\text{or }\;\;\;\;
\begin{minipage}{2.25cm}
\begin{tikzpicture}[yscale=1.001,xscale=1.001]
 


\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
--++(45:1*0.3)  coordinate (X3) 
--++(45:1*0.3) 
--++(135:4*0.3)--++(-135:4*0.3)--++(-45:1*0.3)  coordinate (Y3) 
--++(-45:1*0.3)  coordinate (Y2) 
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;  
 \draw[densely dotted] (X1)--++(135:0.3*4);
  \draw[densely dotted] (X2)--++(135:0.3*4);
    \draw[densely dotted] (X3)--++(135:0.3*4);
      \draw[densely dotted] (Y1)--++(45:0.3*4);
    \draw[densely dotted] (Y2)--++(45:0.3*4);
        \draw[densely dotted] (Y3)--++(45:0.3*4);
        
        
         
    
\draw [very thick,fill=white] (L1)--++(45:0.3*4)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)
 --++(-135:0.3)--++(135:0.3)
--++(-135:0.3*2)
 --++(-45:0.3*3);

    \draw [very thick,fill=gray!40] 
          (L1)--++(45:0.3*3)--++(135:0.3*1)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*1)--++(135:0.3)--++(-135:0.3*1)
 --++(-45:0.3*3);
             
             
             
             
               \draw[very thick] (Y3)--++(45:0.3*2);               \draw[very thick] (Y2)--++(45:0.3*3);
               \draw[very thick] (Y1)--++(45:0.3*4);               
         \draw[very thick] (X3)--++(135:0.3*2);               \draw[very thick] (X2)--++(135:0.3*3);
               \draw[very thick] (X1)--++(135:0.3*4);               



\path(L1)--++(45:0.5*0.3)--++(135:3.5*0.3) coordinate (start);

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(45:0.3) coordinate (start);

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[magenta] (start) circle (2pt);
 
\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(45:0.3) coordinate (start);

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[magenta] (start) circle (2pt);

%
%\fill[cyan] (start) circle (1.8pt);
 \path (start)--++(45:0.3) coordinate (start);

%\fill[cyan] (start) circle (1.8pt);
\path(start)--++(-45:0.3) coordinate (start);

\fill[orange] (start) circle (2pt);



\draw [very thick] (L1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (L1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);

  \path(L1)--++(-90:0.21) node {};


\end{tikzpicture}
\end{minipage}
\;\;\text{or }\;\;\;\;
\begin{minipage}{2.25cm}
\begin{tikzpicture}[yscale=1.001,xscale=-1.001]
 


\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
--++(45:1*0.3)  coordinate (X3) 
--++(45:1*0.3) 
--++(135:4*0.3)--++(-135:4*0.3)--++(-45:1*0.3)  coordinate (Y3) 
--++(-45:1*0.3)  coordinate (Y2) 
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;  
 \draw[densely dotted] (X1)--++(135:0.3*4);
  \draw[densely dotted] (X2)--++(135:0.3*4);
    \draw[densely dotted] (X3)--++(135:0.3*4);
      \draw[densely dotted] (Y1)--++(45:0.3*4);
    \draw[densely dotted] (Y2)--++(45:0.3*4);
        \draw[densely dotted] (Y3)--++(45:0.3*4);
        
        
         
    
\draw [very thick,fill=white] (L1)--++(45:0.3*4)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)
 --++(-135:0.3)--++(135:0.3)
--++(-135:0.3*2)
 --++(-45:0.3*3);

    \draw [very thick,fill=gray!40] 
          (L1)--++(45:0.3*3)--++(135:0.3*1)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*1)--++(135:0.3)--++(-135:0.3*1)
 --++(-45:0.3*3);
             
             
             
             
               \draw[very thick] (Y3)--++(45:0.3*2);               \draw[very thick] (Y2)--++(45:0.3*3);
               \draw[very thick] (Y1)--++(45:0.3*4);               
         \draw[very thick] (X3)--++(135:0.3*2);               \draw[very thick] (X2)--++(135:0.3*3);
               \draw[very thick] (X1)--++(135:0.3*4);               



\path(L1)--++(45:0.5*0.3)--++(135:3.5*0.3) coordinate (start);

\fill[orange] (start) circle (1.8pt);
\draw[orange,very thick] (start)--++(45:0.3) coordinate (start);

\fill[orange] (start) circle (1.8pt);
\draw[orange,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[orange] (start) circle (2pt);


 \path(start)--++(45:0.3) coordinate (start);

 \path(start) circle (1.8pt);
 \path (start)--++(-45:0.3) coordinate (start);
 

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(45:0.3) coordinate (start);

\fill[magenta] (start) circle (1.8pt);
\draw[magenta,very thick] (start)--++(-45:0.3) coordinate (start);

\fill[magenta] (start) circle (2pt);



\draw [very thick] (L1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (L1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);

  \path(L1)--++(-90:0.21) node {};


\end{tikzpicture}
\end{minipage}$$%$$
%\begin{minipage}{1.75cm}
%\begin{tikzpicture}[yscale=1.001,xscale=-1.001]
% 
%
%
%\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
%\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
%\draw [very thick,fill=white] (L1)--++(45:0.3*3)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*2)
% --++(-45:0.3*3);
%
%    \draw [very thick,fill=gray!40] (L1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
% 
%
%
%\path(L1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%
%\fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (2pt);
% 
%\fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (2pt);
%
%
%
%\draw [very thick] (L1)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
%--++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (L1)
%--++(45:0.6)--++(135:0.3)--++(45:0.3);;
%
%\draw [very thick] (L1)
%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
%  \path(L1)--++(-90:0.21) node {$\SSTT$};
%
%
%\end{tikzpicture}
%\end{minipage}
%\xrightarrow {\ \ \ \ \ }
%\begin{minipage}{1.75cm}
%\begin{tikzpicture}[yscale=1.2,xscale=1.2]
% 
%
%
%\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
%\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
%\draw [very thick,fill=white] (L1)--++(45:0.3*3)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*2)
% --++(-45:0.3*3);
%
%    \draw [very thick,fill=gray!40] (L1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
% 
%
%
%\path(L1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%
%\fill[darkgreen] (start) circle (1.8pt);
%\draw[darkgreen,very thick] (start)--++(45:0.3) coordinate (start);
%
%\fill[darkgreen] (start) circle (1.8pt);
%\draw[darkgreen,very thick] (start)--++(-45:0.3) coordinate (start);
%
%\fill[darkgreen] (start) circle (2pt);
% 
%\fill[darkgreen] (start) circle (1.8pt);
%\path (start)--++(45:0.3) coordinate (start);
%
%%\fill[cyan] (start) circle (1.8pt);
%\path (start)--++(-45:0.3) coordinate (start);
%
%\fill[magenta] (start) circle (2pt);
%
%
%
%\draw [very thick] (L1)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
%--++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (L1)
%--++(45:0.6)--++(135:0.3)--++(45:0.3);;
%
%\draw [very thick] (L1)
%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
% \path(L1)--++(-90:0.21) node {$\SSTT$};
%
%\end{tikzpicture}
%\end{minipage} 
%\quad\text {or} \quad
%\begin{minipage}{1.75cm}
%\begin{tikzpicture}[yscale=1.001,xscale=-1.001]
% 
%
%
%\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
%\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
%\draw [very thick,fill=white] (L1)--++(45:0.3*3)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*2)
% --++(-45:0.3*3);
%
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%
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%
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%
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%
%\fill[darkgreen] (start) circle (2pt);
% 
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%\path (start)--++(45:0.3) coordinate (start);
%
%%\fill[cyan] (start) circle (1.8pt);
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%
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%
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%
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%
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%
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%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
% \path(L1)--++(-90:0.21) node {$\SSTT$};
%
%\end{tikzpicture}
%\end{minipage}$$

  
\end{itemize} 
We extend this to a partial ordering, $\prec$, by taking the transitive closure of $\to $.  \end{defn}
 
      




\begin{figure}[ht!]
$$
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%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
%  
%\draw [very thick,fill=white] (LLL1)--++(45:0.3*3)--++(135:0.3)--++(-135:0.3) --++(135:0.3*2)--++(-135:0.3*2)
% --++(-45:0.3*3);
%
% 
%
%
%
%    \draw [very thick,fill=gray!40] (LLL1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
%
%\path(LLL1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%
%\fill[magenta] (start) circle (1.8pt);
%\draw[magenta,very thick] (start)--++(45:0.3) coordinate (start);
%
%\fill[magenta] (start) circle (1.8pt);
%\draw[magenta,very thick] (start)--++(-45:0.3) coordinate (start);
%
%\fill[magenta] (start) circle (2pt);
% 
%
%\path (start)--++(45:0.3) coordinate (start);
%
%
%\path (start)--++(-45:0.3) coordinate (start);
%
%\fill[brown] (start) circle (2pt);
%
%
%\draw [very thick] (LLL1)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
%--++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (LLL1)
%--++(45:0.6)--++(135:0.3)--++(45:0.3);;
%
%\draw [very thick] (LLL1)
%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
%
%
%
%
%
%
%
%\path(-3,-6.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(LL1);
%\path(LL1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (LL1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%  \draw [very thick,fill=white] (LL1)--++(45:0.3*2)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*3);
% 
%
%
%
%
%    \draw [very thick,fill=gray!40] (LL1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
%
%\path(LL1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%
%\fill[darkgreen] (start) circle (1.8pt);
%\path(start)--++(45:0.3) coordinate (start);
%
% 
%\path (start)--++(-45:0.3) coordinate (start);
%
%\fill[brown] (start) circle (2pt);
%  
%
%
%
%
%
%\draw [very thick] (LL1)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
%--++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (LL1)
%--++(45:0.6)--++(135:0.3);
%
%\draw [very thick] (LL1)
%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
%
%
%
%
%
%
%
%
%
%
%
%
%\path(-1,-6.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
%\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
%\draw [very thick,fill=white] (L1)--++(45:0.3*3)--++(135:0.3*3) --++(-135:0.3*3)
% --++(-45:0.3*3);
%
%
%
%
%
%
%    \draw [very thick,fill=gray!40] (L1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
%
%\path(L1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%
%\fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (2pt);
%
%  \fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (1.8pt);
%\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);
%
%\fill[cyan] (start) circle (2pt);
%
% 
%\path (start)--++(-135:0.3)--++(135:0.3) coordinate (start);
%\fill[brown] (start) circle (2pt);
%
%%
%%
%%
%%
%%\path(L1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%%
%%\fill[cyan] (start) circle (1.8pt);
%%\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);
%%
%% 
%%\fill[cyan] (start) circle (2pt);
%%
%%
%%\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);
%%
%%\fill[cyan] (start) circle (1.8pt);
%%\draw[cyan,very thick] (start)--++(45:0.3) coordinate (start);
%%
%%
%%\fill[cyan] (start) circle (1.8pt);
%%\draw[cyan,very thick] (start)--++(-45:0.3) coordinate (start);
%%
%%\fill[cyan] (start) circle (2pt);
%%
%% 
%%\path (start)--++( 135:0.3)--++(135:0.3) coordinate (start);
%%\fill[brown] (start) circle (2pt);
%
%
%
%\draw [very thick] (L1)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
%--++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (L1)
%--++(45:0.6)--++(135:0.3)--++(45:0.3);;
%
%\draw [very thick] (L1)
%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
%
%
%
%
%
%
% 
%\path(1,-6.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(R1);
%\path(R1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (R1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
%\draw [very thick,fill=white] (R1)--++(45:0.3*3)--++(135:0.3)   --++(-135:0.3*2)
%--++(135:0.3*2)
%--++(-135:0.3)
% --++(-45:0.3*3);
%
%
%
%
%    \draw [very thick,fill=gray!40] (R1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
%
%\path(R1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%
%\fill[darkgreen] (start) circle (1.8pt);
%  
%\path (start)--++(-45:0.3*2)--++(45:0.3*2) coordinate (start);
%\fill[lime!80!black] (start) circle (2pt);
%
%
%
%\draw [very thick] (R1)
%--++(45:0.3)--++(135:0.3)
% --++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (R1)
%--++(45:0.6)--++(135:0.3)--++(45:0.3);;
%
%\draw [very thick] (R1)
%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
%
%
%
%
% 
%\path(3,-6.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(RR1);
%\path(RR1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (RR1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
% 
%
%\draw [very thick,fill=white] (RR1)--++(45:0.3*3)--++(135:0.3)--++(-135:0.3*1)--++(135:0.3)--++(-135:0.3*2)
% --++(-45:0.3*2);
%
%
%
%
%
%
%
%    \draw [very thick,fill=gray!40] (RR1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
%
%\path(RR1)--++(45:1.5*0.3)--++(135:1.5*0.3) coordinate (start);
%
%\fill[brown] (start) circle (1.8pt);
%  
%\path (start)--++(-45:0.3*1)--++(45:0.3*1) coordinate (start);
%\fill[lime!80!black] (start) circle (2pt);
%
%
% 
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%\draw [very thick] (RR1)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
% --++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (RR1)
%--++(45:0.6)--++(135:0.3)--++(45:0.3);;
%
% 
%
%
%
%
%
%
%
%
%
%
% 
%\path(5,-6.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(RRR1);
%\path(RRR1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
%\fill[white](circle) circle (20pt);
%
%\draw[densely dotted] (RRR1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2) 
%--++(45:1*0.3)  
%--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2) 
%--++(-45:1*0.3)  coordinate (Y1)
%--++(-45:1*0.3)    ;  
% \draw[densely dotted] (X1)--++(135:0.3*3);
%  \draw[densely dotted] (X2)--++(135:0.3*3);
%  \draw[densely dotted] (Y1)--++(45:0.3*3);
%    \draw[densely dotted] (Y2)--++(45:0.3*3);
%    
% 
%\draw [very thick,fill=white] (RRR1)--++(45:0.3*3)--++(135:0.3*2)  --++(-135:0.3*2)
%--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*3);
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%    \draw [very thick,fill=gray!40] (RRR1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
% --++(-45:0.3*2);
%
%\path(RRR1)--++(45:0.5*0.3)--++(135:2.5*0.3) coordinate (start);
%
%\fill[magenta] (start) circle (1.8pt);
%\path (start)--++(45:0.3) coordinate (start);
%
% \path (start)--++(-45:0.3) coordinate (start);
%
%\fill[violet] (start) circle (2pt);
%
%  \fill[violet] (start) circle (1.8pt);
%\draw[violet,very thick] (start)--++(45:0.3) coordinate (start);
%
%\fill[violet] (start) circle (1.8pt);
%\draw[violet,very thick] (start)--++(-45:0.3) coordinate (start);
%
%\fill[violet] (start) circle (2pt);
%
%  
%
%
%
%\draw [very thick] (RRR1)
%--++(45:0.3)--++(135:0.3)
%--++(45:0.3)--++(135:0.3)
% --++(-135:0.3)--++(-45:0.3)
%--++(-135:0.3)--++(-45:0.3);
%
%
%
%\draw [very thick] (RRR1)
%--++(45:0.6)--++(135:0.3)--++(45:0.3);;
%
%\draw [very thick] (RRR1)
%--++(135:0.6)--++(45:0.3)--++(135:0.3);
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%\end{tikzpicture}
\begin{tikzpicture}[scale=0.888 ]%[scale=0.925 ]


 \draw[very thick] (0,0)--(-5,-2.5)
  (0,0)--(-3,-2.5)
   (0,0)--(-1,-2.5)
    (0,0)--(1,-2.5)
     (0,0)--(3,-2.5)
      (0,0)--(5,-2.5);




 \draw[very thick]
   (-5,-2.5)--(-5,-6.5)
      (-5,-2.5)--(-3,-6.5)
   (-3,-2.5)--(1,-6.5)
      (-3,-2.5)--(5,-6.5)
(-3,-6.5)--(-3,-2.5);



 \draw[very thick]
   (-1,-2.5)--(-5,-6.5)
   (-1,-2.5)--(5,-6.5)    (-1,-2.5)--(-1,-6.5) ;




 \draw[very thick]
   (1,-2.5)--(-1,-6.5)
   (1,-2.5)--(-3,-6.5)    (1,-2.5)--(3,-6.5) ;









 \draw[very thick]
   (5,-2.5)--(5,-6.5)
      (5,-2.5)--(3,-6.5)
   (3,-2.5)--(1,-6.5)
      (3,-2.5)--(3,-6.5)
      (3,-2.5)--(-5,-6.5)   ;










 \draw[very thick] (0,-6.5-2.5)--(-5,-6.5)
  (0,-6.5-2.5)--(-3,-6.5)
   (0,-6.5-2.5)--(-1,-6.5)
    (0,-6.5-2.5)--(1,-6.5)
     (0,-6.5-2.5)--(3,-6.5)
      (0,-6.5-2.5)--(5,-6.5);



\path(0,0)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(top);
\path(top)--++(45:0.3*1.5)--++(135:0.3*1.5)  coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (top)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);

\begin{scope}


\draw [very thick] (top)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
 --++(-45:0.3*2)--++(45:0.2 );


\clip(top)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
 --++(-45:0.3*2);


\draw [very thick,fill=green!80!black] (top)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (top)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (top)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\end{scope}


\draw [very thick] (top)--++(45:0.3)--++(135:0.3)--++(-135:0.3);



\path(0,-6.5-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(bottom);
\path(bottom)--++(45:0.3*1.5)--++(135:0.3*1.5)  coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (bottom)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);

\draw [very thick,fill=gray!40] (bottom)--++(45:0.3*3)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)
 --++(-45:0.3*3);




\begin{scope}


\draw [very thick] (bottom)--++(45:0.3*3)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3);



\clip(bottom)--++(45:0.3*3)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3)--++(135:0.3)--++(-135:0.3);


\draw [very thick,fill=orange] (bottom)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (bottom)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (bottom)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (bottom)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (bottom)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}




\draw [very thick] (bottom)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (bottom)
--++(45:0.6)--++(135:0.3);

\draw [very thick] (bottom)
--++(135:0.6)--++(45:0.3);






















\path(-5,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(LLL1);
\path(LLL1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (LLL1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);




\draw [very thick,fill=gray!40] (LLL1)--++(45:0.3*2)--++(135:0.3) --++(135:0.3*2)--++(-135:0.3*2)
 --++(-45:0.3*3);


 \begin{scope}


\draw [very thick] (LLL1)--++(45:0.3*2)--++(135:0.3) --++(135:0.3*2)--++(-135:0.3*2)
 --++(-45:0.3*3);




\clip(LLL1)--++(45:0.3*2)--++(135:0.3) --++(135:0.3*2)--++(-135:0.3*2)
 --++(-45:0.3*3);



\draw [very thick,fill=orange] (LLL1)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (LLL1)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (LLL1)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (LLL1)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (LLL1)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}


\draw [very thick] (LLL1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (LLL1)
--++(45:0.6)--++(135:0.3);

\draw [very thick] (LLL1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);











\path(-3,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(LL1);
\path(LL1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (LL1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);





\draw [very thick,fill=gray!40] (LL1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3*2)--++(-135:0.3)
 --++(-45:0.3*3);


 \begin{scope}


\draw [very thick] (LL1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3*2)--++(-135:0.3)
 --++(-45:0.3*3);




\clip(LL1)--++(45:0.3*2)--++(135:0.3)--++(-135:0.3)--++(135:0.3*2)--++(-135:0.3)
 --++(-45:0.3*3);



\draw [very thick,fill=orange] (LL1)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (LL1)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (LL1)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (LL1)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (LL1)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}






\draw [very thick] (LL1)
--++(45:0.3)--++(135:0.3)
 --++(-135:0.3)--++(-45:0.3);




\draw [very thick] (LL1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);












\path(-1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(L1);
\path(L1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (L1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);

\draw [very thick,fill=gray!40] (L1)--++(45:0.3*3)--++(135:0.3*2)--++(-135:0.3)--++(135:0.3)--++(-135:0.3*2)
 --++(-45:0.3*3);






\draw [very thick] (L1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);


 \begin{scope}


\draw [very thick] (L1)
--++(45:0.9)--++(135:0.6)
--++(-135:0.3)--++(135:0.3)
--++(-135:0.6)--++(-45:0.9);




\clip(L1)
(L1)
--++(45:0.9)--++(135:0.6)
--++(-135:0.3)--++(135:0.3)
--++(-135:0.6)--++(-45:0.9);




\draw [very thick,fill=orange] (L1)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (L1)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (L1)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (L1)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (L1)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}



\draw [very thick] (L1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);
















\path(1,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(R1);
\path(R1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (R1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);

\draw [very thick,fill=gray!40] (R1)--++(45:0.3*2)--++(135:0.3) --++(135:0.3)--++(-135:0.3*2)
 --++(-45:0.3*2);





 \begin{scope}


\draw [very thick] (R1)
--++(45:0.3*2)--++(135:0.3) --++(135:0.3)--++(-135:0.3*2)
 --++(-45:0.3*2);




\clip(R1)
(R1)
--++(45:0.3*2)--++(135:0.3) --++(135:0.3)--++(-135:0.3*2)
 --++(-45:0.3*2);



\draw [very thick,fill=orange] (R1)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (R1)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (R1)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (R1)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (R1)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}






\draw [very thick] (R1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);

















\draw [very thick] (L1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (L1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);










\path(3,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(RR1);
\path(RR1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (RR1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);

 \draw [very thick,fill=gray!40] (RR1)--++(45:0.3*3)--++(135:0.3)--++(-135:0.3*2)--++(135:0.3)--++(-135:0.3)
 --++(-45:0.3*2);






 \begin{scope}


\draw [very thick] (RR1)
--++(45:0.3*3)--++(135:0.3)--++(-135:0.3*2)--++(135:0.3)--++(-135:0.3)
 --++(-45:0.3*2);




\clip(RR1)
(RR1)
--++(45:0.3*3)--++(135:0.3)--++(-135:0.3*2)--++(135:0.3)--++(-135:0.3)
 --++(-45:0.3*2);



\draw [very thick,fill=orange] (RR1)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (RR1)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (RR1)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (RR1)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (RR1)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}











\draw [very thick] (RR1)
--++(45:0.3)--++(135:0.3)
 --++(-135:0.3)--++(-45:0.3);



\draw [very thick] (RR1)
--++(45:0.6)--++(135:0.3);






\path(5,-2.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(RRR1);
\path(RRR1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (RRR1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);

 \draw [very thick,fill=gray!40] (RRR1)--++(45:0.3*3)--++(135:0.3*2)  --++(-135:0.3*3)
 --++(-45:0.3*2);





 \begin{scope}


\draw [very thick] (RRR1)
--++(45:0.3*3)--++(135:0.3*2)  --++(-135:0.3*3)
 --++(-45:0.3*2);




\clip(RRR1)
(RRR1)
--++(45:0.3*3)--++(135:0.3*2)  --++(-135:0.3*3)
 --++(-45:0.3*2);



\draw [very thick,fill=orange] (RRR1)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (RRR1)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (RRR1)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (RRR1)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (RRR1)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}








\draw [very thick] (RRR1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (RRR1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

































































\path(-5,-6.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(LLL1);
\path(LLL1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

\draw[densely dotted] (LLL1)--++(45:1*0.3) coordinate (X1)--++(45:1*0.3) coordinate (X2)
--++(45:1*0.3)
--++(135:3*0.3)--++(-135:3*0.3)--++(-45:1*0.3)  coordinate (Y2)
--++(-45:1*0.3)  coordinate (Y1)
--++(-45:1*0.3)    ;
 \draw[densely dotted] (X1)--++(135:0.3*3);
  \draw[densely dotted] (X2)--++(135:0.3*3);
  \draw[densely dotted] (Y1)--++(45:0.3*3);
    \draw[densely dotted] (Y2)--++(45:0.3*3);


\draw [very thick,fill=gray!40] (LLL1)--++(45:0.3*3)--++(135:0.3)--++(-135:0.3) --++(135:0.3*2)--++(-135:0.3*2)
 --++(-45:0.3*3);




 \begin{scope}


\draw [very thick] (LLL1)
--++(45:0.3*3)--++(135:0.3)--++(-135:0.3) --++(135:0.3*2)--++(-135:0.3*2)
 --++(-45:0.3*3);




\clip(LLL1)
(LLL1)
--++(45:0.3*3)--++(135:0.3)--++(-135:0.3) --++(135:0.3*2)--++(-135:0.3*2)
 --++(-45:0.3*3);



\draw [very thick,fill=orange] (LLL1)--++(45:0.3 )--++(135:0.9)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);

\draw [very thick,fill=green!80!black] (LLL1)--++(45:0.3 )--++(135:0.6)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=magenta] (LLL1)--++(45:0.3 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=cyan] (LLL1)--++(45:0.6 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\draw [very thick,fill=violet!80] (LLL1)--++(45:0.9 )--++(135:0.3)--++(45:0.3) --++(135:0.3)
--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
\end{scope}






\draw [very thick] (LLL1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
--++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (LLL1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (LLL1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);








\path(-3,-6.5)--++(-45:0.3*1.5)--++(-135:0.3*1.5) coordinate(LL1);
\path(LL1)--++(45:0.3*1.5)--++(135:0.3*1.5) coordinate (circle);
\fill[white](circle) circle (20pt);

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 \begin{scope}


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(LL1)
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\draw [very thick] (L1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);








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 \begin{scope}


\draw [very thick] (R1)
--++(45:0.3*3)--++(135:0.3)   --++(-135:0.3*2)
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--++(-135:0.3)
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(R1)
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\draw [very thick] (R1)
--++(45:0.3)--++(135:0.3)
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\draw [very thick] (R1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (R1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);






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 \begin{scope}


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--++(-135:0.3)--++(-45:0.3)--++(-135:0.3)--++(-45:0.3);
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\draw [very thick] (RR1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
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--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (RR1)
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 \begin{scope}


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(RRR1)
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\draw [very thick] (RRR1)
--++(45:0.3)--++(135:0.3)
--++(45:0.3)--++(135:0.3)
 --++(-135:0.3)--++(-45:0.3)
--++(-135:0.3)--++(-45:0.3);



\draw [very thick] (RRR1)
--++(45:0.6)--++(135:0.3)--++(45:0.3);;

\draw [very thick] (RRR1)
--++(135:0.6)--++(45:0.3)--++(135:0.3);




















































\end{tikzpicture} $$
\caption{The submodule lattice  of  $\Delta^\Bbbk (2,1)$ for $m=n=3$.  %Grey/black  lines    indicate partitions which differ  by adding/splitting a Dyck path.
%The 0th level is at the top and the 3rd level is at the bottom (following the convention that submodules appear at the ``bottom" of a diagram). 
}\label{standard21}
\end{figure}
 
 An example of the lattice on $\Delta^\Bbbk(\la)$ for $\la=(2,1)$ and $m=n=3$ is depicted in \cref{standard21}.
With a little more work, one can prove that there is a unique Dyck pair $(\la, \alpha)$ of maximal degree (and that the submodule lattice is a bonafide lattice in the combinatorial sense!).  Indeed we have the following:


\begin{thm}[B.-D.-H.-S. \cite{compan2}]
Fix $\la \in {^PW}$ for  $(W,P)=(\mathfrak{S}_{n+m},\mathfrak{S}_{n}\times \mathfrak{S}_{m})$. 
The module $\Delta^\Bbbk(\la)$ has a unique simple submodule, it is rigid (its socle and radical layers coincide) and 
the full submodule lattice of $\Delta(\la)$ is given by the partial ordering $\prec$.
 

\end{thm}
  
  \begin{proof}
We first provide a full quiver and relations presentation of $\mathscr{H}_{(W,P)}  $ and then use this to analyse the submodule structures of $\Delta^\Bbbk(\la)$.
For example let $\mu=(3^2,1)$ as in the leftmost vertex of the penultimate layer of the module of the module $\Delta^\Bbbk(2,1)$ depicted in \cref{standard21}. The composition factor $L^\Bbbk(3^2,1)$  has three distinct paths leading into it; these  come from the simple modules labelled by 
$(3^2)$, $(3^2,2)$, and $(2,1^2)$ respectively.   
These three paths can be seen to be equal using the fork-spot relations as follows:
    $$
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 \end{tikzpicture} 
 \end{minipage}=\;\;\;\begin{minipage}{4.75cm}\begin{tikzpicture}[scale=1.1]

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 \end{tikzpicture} 
 \end{minipage}=\;\;\;
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\clip(-0.25,-1.5) rectangle (3.75,1.5);


\draw[magenta,line width=0.08cm](0,-1.5)--++(90:4.5+1.5);

\draw[darkgreen,line width=0.08cm](0.5,0-1.5)--++(90:3.5+1.5) coordinate (X);
  \fill[darkgreen ] (X) circle (3pt);
  \draw[cyan,line width=0.08cm](1,0-1.5) --++(90:2+1.5) coordinate (X);
   \fill[cyan ] (X) circle (3pt);

 \draw[orange,line width=0.08cm](1.5,-1.5)  --++(90:0.5) coordinate (X);
 \fill[orange ] (X) circle (3pt);
 \draw[magenta,line width=0.08cm](2,-1.5)  --++(90:0.5) coordinate (X);
  \fill[magenta ] (X) circle (3pt);
  \draw[violet,line width=0.08cm](2.5,-1.5)  --++(90:1.85) coordinate (X);
  \fill[violet ] (X) circle (3pt);

  \draw[darkgreen,line width=0.08cm] (0.5,-0.5)--++(0:1.2) to [out=0,in=90]  (1.5+1.5,-1)--++(90:-1.5);
%  \draw[cyan,line width=0.08cm] (1,1.1)--++(0:1.2) to [out=0,in=90]  (2+1.5,0);

 



 

 \end{tikzpicture} 
 \end{minipage}$$
The remaining cases also follow by fork-spot relations, but with a little more thought required.  One must then show that these 
  relations are exhaustive --- this requires the full quiver and relations presentation of  $\mathscr{H}_{(W,P)}  $ alluded to above.
  \end{proof}
  
  
\printbibliography


  \end{document}