
% PRELUDE

\begin{figure}[htbp]
%\includegraphics[width=0.8\textwidth]{a2pic.png}
\begin{center}




% TYPE A_2 


\resizebox{.32\textwidth}{!}{
\begin{tikzpicture}[scale =1,bull/.style={circle,draw=black,fill=black,thick,scale=.6}]
\tikzmath{\xpoint=1.732050808;\ypoint=1;\xbdry=1.154700538;\ybdry=1;\hght=8;}
\clip (.8660254038,.5) circle (3.1);
% \clip(0,0)circle(2.5);
\newcounter{co};
\newcounter{li};
\newcounter{xcurr};
\newcounter{ycurr};
%draws a hexagon centered around point (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)
% \setcounter{xcurr}{0};
% \setcounter{ycurr}{0};
% \node at (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)[bull]{};
% \draw[-,black,very thin](\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--cycle;

%Inner loop iterates vertically
%outer loop iterates horizontally
\setcounter{li}{-2}
\whiledo{\value{li}<3}
{
    \setcounter{co}{-3}
    \whiledo{\value{co}<4}
    {
        \setcounter{xcurr}{2*\value{li}};
        \setcounter{ycurr}{2*\value{co}};
        \node at (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)[bull]{};
        \draw[-,black,very thin](\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--cycle;
        \stepcounter{co};
    }
    \stepcounter{li};
}

\setcounter{li}{-2};
\whiledo{\value{li}<2}
{
    \setcounter{co}{-4};
    \whiledo{\value{co}<4}
    {
        \setcounter{xcurr}{2*\value{li}+1};
        \setcounter{ycurr}{2*\value{co}+1};
        \node at (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)[bull]{};
        \draw[-,black,very thin](\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--cycle;
        \stepcounter{co};
    }
    \stepcounter{li};
}
%time for more loops to iterate the dominant region nice
\setcounter{co}{0}
\whiledo{\value{co}<8}
{
    \draw[-,black,thick] (\value{co}*\xbdry,0)--(\value{co}*\xbdry+.5*\hght*\xbdry,\hght*\ybdry);
    
    \draw[-,black,thick](\value{co}*.5*\xbdry,\value{co}*\ybdry)--(\value{co}*.5*\xbdry+\hght*\xbdry,\value{co}*\ybdry);
    
    \draw[-,black,thick](\value{co}*.5*\xbdry,\value{co}*\ybdry)--(\value{co}*\xbdry,0);
    
    \stepcounter{co};
}
%time for some c++ generated lines of code eh?pls dont pay attention too much to this its going to be fairly manual adjustments, but if you wanna know its going by rows of alcoves oriented the same direction
%1st row
\node at (.5*\xbdry+0*\xbdry,1/3){\scalebox{1}{$\emptyset$}};
\node at (.5*\xbdry+1*\xbdry,1/3){\scalebox{.4}{\yng(2)}};
\node at (.5*\xbdry+2*\xbdry,1/3){\scalebox{.25}{\yng(4,2)}};
\node at (.5*\xbdry+3*\xbdry,1/3){\scalebox{.18}{\yng(6,4,2)}};
%2nd row
\node at (\xbdry+0*\xbdry,2/3){\scalebox{.4}{\yng(1)}};
\node at (\xbdry+1*\xbdry,2/3){\scalebox{.25}{\yng(3,1)}};
\node at (\xbdry+2*\xbdry,2/3){\scalebox{.18}{\yng(5,3,1)}};
\node at (\xbdry+3*\xbdry,2/3){\scalebox{.12}{\yng(7,5,3,1)}};

%3rd row
\node at (\xbdry+0*\xbdry,4/3){\scalebox{.4}{\yng(1,1)}};
\node at (\xbdry+1*\xbdry,4/3){\scalebox{.25}{\yng(3,1,1)}};
\node at (\xbdry+2*\xbdry,4/3){\scalebox{.18}{\yng(5,3,1,1)}};
%4th row 
\node at (1.5*\xbdry+0*\xbdry,5/3){\scalebox{.25}{\yng(2,1,1)}};
\node at (1.5*\xbdry+1*\xbdry,5/3){\scalebox{.18}{\yng(4,2,1,1)}};
\node at (1.5*\xbdry+2*\xbdry,5/3){\scalebox{.12}{\yng(6,4,2,1,1)}};
%5th row
\node at (1.5*\xbdry+0*\xbdry,7/3){\scalebox{.25}{\yng(2,2,1,1)}};
\node at (1.5*\xbdry+1*\xbdry,7/3){\scalebox{.18}{\yng(4,2,2,1,1)}};
\node at (1.5*\xbdry+2*\xbdry,7/3){\scalebox{.12}{\yng(6,4,2,2,1,1)}};
%6th row
\node at (2*\xbdry+0*\xbdry,8/3){\scalebox{.18}{\yng(3,2,2,1,1)}};
\node at (2*\xbdry+1*\xbdry,8/3){\scalebox{.12}{\yng(5,3,2,2,1,1)}};
%7th row
\node at (2*\xbdry+0*\xbdry,10/3){\scalebox{.18}{\yng(3,3,2,2,1,1)}};
%8th row
\node at (2.5*\xbdry+0*\xbdry,11/3){\scalebox{.12}{\yng(4,3,2,2,1,1)}};
\end{tikzpicture}}
%\qquad\qquad
%
%
%
%
% TYPE C_2 
%
%
%
%
\resizebox{.32\textwidth}{!}{
\begin{tikzpicture}[scale = 1,bull/.style={circle,draw=black,fill=black,thick,scale=.6}]
\tikzmath{\xpoint=1.732050808;\ypoint=1;\xbdry=1.154700538;\ybdry=1;\hght=8;\smallx=.8660254038;}
\clip (.8660254038,.5) circle (3.1);

%draws a hexagon centered around point (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)
% \setcounter{xcurr}{0};
% \setcounter{ycurr}{0};
% \node at (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)[bull]{};
% \draw[-,black,very thin](\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--cycle;

%Inner loop iterates vertically
%outer loop iterates horizontally
\setcounter{li}{-2}
\whiledo{\value{li}<3}
{
    \setcounter{co}{-3}
    \whiledo{\value{co}<4}
    {
        \setcounter{xcurr}{2*\value{li}};
        \setcounter{ycurr}{2*\value{co}};
        \node at (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)[bull]{};
        \draw[-,black,very thin](\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--cycle;
        \stepcounter{co};
    }
    \stepcounter{li};
}

\setcounter{li}{-2};
\whiledo{\value{li}<2}
{
    \setcounter{co}{-4};
    \whiledo{\value{co}<4}
    {
        \setcounter{xcurr}{2*\value{li}+1};
        \setcounter{ycurr}{2*\value{co}+1};
        \node at (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)[bull]{};
        \draw[-,black,very thin](\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint)--(\value{xcurr}*\xpoint-.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint+.5*\xbdry,\value{ycurr}*\ypoint-\ybdry)--cycle;
        \stepcounter{co};
    }
    \stepcounter{li};
}
%time for more loops to iterate the dominant region nice
\setcounter{co}{0}
\whiledo{\value{co}<4}
{
    \draw[-,black,thick](1.5*\value{co}*\xbdry,\value{co}*3)--(1.5*\value{co}*\xbdry+6*\xbdry,\value{co}*3+.5*\hght);
    
    %vertical lines
    \draw[-,black,thick](\value{co}*\xpoint,3*\value{co})--(\value{co}*\xpoint,\value{co});
    
    \draw[-,black,thick](\value{co}*\smallx,1.5*\value{co})--(\value{co}*\xpoint,\value{co});
    \stepcounter{co};
}
\setcounter{co}{0}
\whiledo{\value{co}<6}
{
    \draw[-,black,thick](\value{co}*\xpoint,\value{co})--(\value{co}*\xbdry+.5*\hght*\xbdry,\hght);
    %horizontal lines
    \draw[-,black,thick](\value{co}*.5*\xbdry,\value{co})--(1.5*\value{co}*\xbdry,\value{co});
    
    \draw[-,black,thick](\value{co}*.5*\xbdry,\value{co})--(\value{co}*\smallx,.5*\value{co});
    \stepcounter{co};

}
%wasnt quite able to generate these in c++ so it was by hand
\node at (.5/3*\xbdry+\smallx/3,1.5/3){\scalebox{.9}{$\emptyset$}};
\node at ({(2*\xbdry+\smallx)/3},2.5/3){\scalebox{.35}{\yng(1)}};
\node at ({(2*\xbdry+\smallx)/3},3.5/3){\scalebox{.23}{\yng(2)}};
\node at ({(\xbdry+\smallx+2*\smallx)/3},4.5/3){\scalebox{.23}{\yng(1,1)}};
\node at ({(3*\xbdry+\xbdry)/3},5/3+.1){\scalebox{.2}{\yng(2,1,1)}};
\node at ({(5*\xbdry)/3+.04},5/3+.1){\scalebox{.2}{\yng(3,1)}};
\node at ({(3.5*\xbdry+3*\smallx)/3+.1},4.5/3+.03){\scalebox{.2}{\yng(3,1,1)}};
\node at ({(3*\xbdry+\xbdry)/3+.04},7/3-.04){\scalebox{.18}{\yng(2,2,1,1)}};
\node at ({(4*\xbdry+\xbdry)/3+.02},7/3-.11){\scalebox{.16}{\yng(4,2)}};
\node at ({(3.5*\xbdry+3*\smallx)/3+.12},7.5/3-.1){\scalebox{.13}{\yng(5,3,1)}};
\node at ({(5*\xbdry+3*\smallx)/3},5.5/3-.06){\scalebox{.16}{\yng(4,2,1,1)}};
\node at ({(5*\xbdry+3*\smallx)/3-.05},6.5/3){\scalebox{.13}{\yng(5,3,1,1)}};
\node at ({(5.5*\xbdry+3*\smallx)/3-.07},7.5/3+.06){\scalebox{.11}{\yng(4,2,2,1,1)}};
\node at ({(4*\xbdry+3*\smallx)/3+.1},8.5/3-.04){\scalebox{.12}{\yng(3,2,2,1,1)}};
\node at ({(4*\xbdry+3*\smallx)/3+.1},9.5/3+.02){\scalebox{.12}{\yng(3,3,2,2,1,1)}};
%not the most pleasant picture i have drawn
\end{tikzpicture}}
% TYPE G_2
\resizebox{.32\textwidth}{!}{\begin{tikzpicture}[scale = 1,bull/.style={circle,draw=black,fill=black,thick,scale=.6}]
\tikzmath{\xpoint=1;\ypoint=1;\xbdry=1;\ybdry=1;\hght=8;\smallx=1; \smally=1;}
\clip (.8660254038,.5) circle (3.1);

%Inner loop iterates vertically
%outer loop iterates horizontally
\setcounter{li}{-2}
\whiledo{\value{li}<3}
{
    \setcounter{co}{-3}
    \whiledo{\value{co}<4}
    {
        \setcounter{xcurr}{2*\value{li}};
        \setcounter{ycurr}{2*\value{co}};
        \node at (\value{xcurr}*\xpoint,\value{ycurr}*\ypoint)[bull]{};
        \draw[-,black,very thin](\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint+\ybdry)--(\value{xcurr}*\xpoint+\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint-\ybdry)--(\value{xcurr}*\xpoint-\xbdry,\value{ycurr}*\ypoint+\ybdry)--cycle;
        \stepcounter{co};
    }
    \stepcounter{li};
}

\setcounter{li}{0}
\whiledo{\value{li}<3}
{
    \setcounter{co}{-1}
    \whiledo{\value{co}<\value{li}}
    {
        \setcounter{xcurr}{2*\value{li}};
        \setcounter{ycurr}{2*\value{co}+2};
        
        % vertical lines
        \draw[-,black,thick] (\value{xcurr}*\xpoint+1*\xpoint,\value{ycurr}*\ypoint+0*\xpoint) -- (\value{xcurr}*\xpoint+1*\xpoint,\value{ycurr}*\ypoint+1*\ypoint);
        \draw[-,black,thick] (\value{xcurr}*\xpoint+2*\xpoint,\value{ycurr}*\ypoint+0*\xpoint) -- (\value{xcurr}*\xpoint+2*\xpoint,\value{ycurr}*\ypoint+2*\ypoint);
        \draw[black,thick] (\value{xcurr}*\xpoint+3*\xpoint,\value{ycurr}*\ypoint+1*\xpoint) -- (\value{xcurr}*\xpoint+3*\xpoint,\value{ycurr}*\ypoint+2*\ypoint);
        
        % down-diagonal lines
        \draw[-,black,thick] (\value{xcurr}*\xpoint+1*\xpoint,\value{ycurr}*\ypoint+1*\xpoint) -- (\value{xcurr}*\xpoint+2*\xpoint,\value{ycurr}*\ypoint+0*\ypoint);
        \draw[-,black,thick] (\value{xcurr}*\xpoint+2*\xpoint,\value{ycurr}*\ypoint+2*\xpoint) -- (\value{xcurr}*\xpoint+3*\xpoint,\value{ycurr}*\ypoint+1*\ypoint);

        % horizontal lines
        \draw[-,black,thick] (\value{xcurr}*\xpoint+0*\xpoint,\value{ycurr}*\ypoint+0*\xpoint) -- (\value{xcurr}*\xpoint+2*\xpoint,\value{ycurr}*\ypoint+0*\ypoint);
        \draw[-,black,thick] (\value{xcurr}*\xpoint+1*\xpoint,\value{ycurr}*\ypoint+1*\xpoint) -- (\value{xcurr}*\xpoint+3*\xpoint,\value{ycurr}*\ypoint+1*\ypoint);
        
        % up-diagonal lines
        \draw[-,black,thick] (\value{xcurr}*\xpoint+0*\xpoint,\value{ycurr}*\ypoint+0*\xpoint) -- (\value{xcurr}*\xpoint+2*\xpoint,\value{ycurr}*\ypoint+2*\ypoint);
        
        \stepcounter{co};
    }
    \stepcounter{li};
}

% just did these by hand %you monster
\node at (0*\xbdry+2*\smallx/3,0*\ybdry+1*\smally/3){\scalebox{.9}{$\emptyset$}};
\node at (1*\xbdry+1*\smallx/3,0*\ybdry+1*\smally/3){\scalebox{.35}{\yng(1)}};
\node at (1*\xbdry+2*\smallx/3,0*\ybdry+2.25*\smally/3){\scalebox{.3}{\yng(2,1)}};
\node at (2*\xbdry+1*\smallx/3,0*\ybdry+2*\smally/3){\scalebox{.23}{\yng(3,1,1)}};
\node at (2*\xbdry+2.25*\smallx/3,0*\ybdry+0.75*\smally/3){\scalebox{.18}{\yng(4,1,1,1)}};
\node at (1*\xbdry+2*\smallx/3,1*\ybdry+1*\smally/3){\scalebox{.3}{\yng(2,2)}};
\node at (2*\xbdry+0.75*\smallx/3,1*\ybdry+0.75*\smally/3){\scalebox{.23}{\yng(3,2,1)}};
\node at (2*\xbdry+2*\smallx/3,1*\ybdry+2.25*\smally/3){\scalebox{.18}{\yng(4,3,2,1)}};
\node at (3*\xbdry+1*\smallx/3,0*\ybdry+0.75*\smally/3){\scalebox{.15}{\yng(5,2,1,1,1)}};
\node at (3*\xbdry+2.25*\smallx/3,0*\ybdry+2.25*\smally/3){\scalebox{.13}{\yng(6,3,2,1,1,1)}};
\node at (3*\xbdry+2.25*\smallx/3,1*\ybdry+0.75*\smally/3){\scalebox{.13}{\yng(6,3,3,1,1,1)}};
\node at (3*\xbdry+1*\smallx/3,1*\ybdry+2*\smally/3){\scalebox{.18}{\yng(5,3,3,1,1)}};
\node at (2*\xbdry+2.25*\smallx/3,2*\ybdry+0.75*\smally/3){\scalebox{.18}{\yng(4,4,2,2)}};
\node at (3*\xbdry+1*\smallx/3,2*\ybdry+0.75*\smally/3){\scalebox{.15}{\yng(5,4,3,2,1)}};
\end{tikzpicture}}




%\includegraphics[width=0.5\textwidth]{c2pic.png}




% POSTLUDE

\end{center}
\caption{$3$-cores in types $A_2$ and $G_2$, and self-conjugate $4$-cores in type $C_2$.}
\label{fig:a2}
\end{figure}








% COMMENTS

\begin{comment}
\begin{tikzpicture}[scale=7]
    % labels
	\tikzmath{\c=2.12132/3;\d=1.22474/3;\a=.707107;\b=1.22474;};
    \draw[gray,-] (\a/3,\b/3) -- (1.41421/3,0) -- (\a/3,-\b/3) -- (-\a/3,-\b/3) -- (-1.41421/3,0) -- (-\a/3,\b/3)--(\a/3,\b/3);
    \draw[gray,-] (\a/3,\b/3) -- (-\a/3,-\b/3);
    \draw[gray,-] (1.41421/3,0) -- (-1.41421/3,0);
    \draw[gray,-] (\a/3,-\b/3) -- (-\a/3,\b/3);
    \draw[gray,-] (\c/2,\d/2) -- (-\c/2,-\d/2);
    \draw[gray,-] (0,2.44949/6) -- (0,-2.44949/6);
    \draw[gray,-] (-\c/2,\d/2) -- (\c/2,-\d/2);
    %\draw[black,->] (0,0) -- (-2*\c*1.5,2*\d*1.5);
    \draw[black,->] (0,0) -- (\a/3,\b/3);
    \draw[black,->] (0,0) -- (\c/2,\d/2);
    %\draw[gray,-] (\c/2,\d/2) -- (\b/3,\a/3);
    \node at (\a/3*1.1,\b/3*1.1) {$\frac{\chk{\lambda_1}}{3}$};
    \node at (\c/2*1.2,\d/2*1.2) {$\frac{\chk{\lambda_2}}{2}$};
    \filldraw (\a/6+\c/6,\b/6+\d/6) circle[radius=0.1pt];
    \filldraw (2.82843/12,2.44949/12) circle[radius=0.1pt];
    \node at (\a/6*1.1+\c/6*1.1,\b/6*1.1+\d/6*1.1) {$\frac{\chk{\rho}}{h}$};
    \node at (2.82843/12*.9,2.44949/12*.9) {$\frac{\rho}{g}$};
 	%\foreach \i in {0,...,#1}{
	%  \foreach \j in {0,...,#2}{
	%	\tikzmath{\l = int(\m*\i-\n*\j);};
	%	\IfStringInList{\l}{#3}{\fill[lightgray] (\m-\j,\i) rectangle (\m+.5,\n+.5);}{};
    %  };
	%};
	%\draw[black,thin] (\m,0) -- (0,\n);
	%\draw[step=1,gray,very thin] (-.5,-.5) grid (\m+.5,\n+.5);
    %\foreach \i in {0,...,#1}{
	%  \foreach \j in {0,...,#2}{
    %	\tikzmath{\l = int(\m*\i-\n*\j);};
	%	\ifthenelse{\l<0}{\tikzmath{\s=int(-1);\l=int(-\l);}}{\tikzmath{\s=int(1);}};
		%\IfStringInList{\l}{#2}{\fill[lightgray] (1,2) rectangle (\m+.5,\n+.5);}{};
		%\pgfmathparse{(#1)};
    %    \path (\m-\j,\i) node{\ifthenelse{\s=-1}{$\overline{\l}$}{$\l$}};
    %  };
	%};
\end{tikzpicture}
\nathan{I think I messed up the normalization here...}
\end{comment}

