\documentclass[11pt]{article}


%\usepackage{showkeys}
\usepackage{amsfonts}
\usepackage{amscd}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath, xspace}
\usepackage{blkarray}
\usepackage{bm}
\usepackage{cancel}
\usepackage{color}
\usepackage{graphics}
\usepackage{graphicx}
\usepackage{enumitem}
\usepackage{fancyhdr}
\pagestyle{fancy}
\usepackage{mathdots}
\usepackage{mathrsfs}
\usepackage{multicol}
\usepackage{stmaryrd}
%\usepackage{youngtab}
\usepackage{ytableau}
%\usepackage{refcheck}
\usepackage[all]{xy}
%\usepackage[neveradjust]{paralist}

\usepackage[plainpages,backref]{hyperref}


\usepackage{lscape}

\def\fiverm{}



\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\theoremstyle{definition}
\newtheorem*{defn}{Definition}	
\newtheorem{remark}[thm]{Remark}
\theoremstyle{example}
\newtheorem{example}{Example}[section]
\newtheorem*{question}{Question}
\newtheorem{conj}[thm]{Conjecture}
\theoremstyle{remark}
\newtheorem*{ack}{Acknowledgments}
\numberwithin{equation}{section}

\setlength{\evensidemargin}{1in} 
\addtolength{\evensidemargin}{-1in}
\setlength{\oddsidemargin}{1in} 
\addtolength{\oddsidemargin}{-1in} 
\setlength{\topmargin}{1in} 
\addtolength{\topmargin}{-1.5in}

\setlength{\textwidth}{17cm} \setlength{\textheight}{23cm}
\setlength{\headwidth}{14cm}
\setlength{\headheight}{13.6pt}

\providecommand{\keywords}[1]{\textbf{\textit{Key words---}} #1}
\newcommand{\btimes}{\mathbin{\rotatebox[origin=c]{90}{$\ltimes$}}}
\newcommand{\utimes}{\mathbin{\rotatebox[origin=c]{-90}{$\ltimes$}}}


%\def\scrC{\mathscr{C}}
%\def\scrCmin{\mathscr{C}_{\mathrm{min}}}
%\def\scrCreg{\mathscr{C}_{\mathrm{reg}}}
%\def\scrCsubreg{\mathscr{C}_{\mathrm{subreg}}}
%\def\ureg{z_{\mathrm{r}}}
%
%
%
%\def\Phis{\Phi_{\mathrm{s}}}
%\def\Phil{\Phi_{\mathrm{l}}}
%


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

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



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


\def\fU{\mathfrak{U}}

\def\fgl{\mathfrak{gl}}
\def\fsl{\mathfrak{sl}}

\def\ad{\mathrm{ad}}
\def\Aut{\mathrm{Aut}}
\def\Card{\mathrm{Card}}
\def\diag{\mathrm{diag}}
\def\dim{\mathrm{dim}}
\def\End{\mathrm{End}}
\def\ev{\mathrm{ev}}
\def\Hom{\mathrm{Hom}}
\def\GL{\mathrm{GL}}
\def\id{\mathrm{id}}
\def\Ind{\mathrm{Ind}}
\def\ind{\mathrm{ind}}
\def\Inf{\mathrm{Inf}}
\def\Irr{\mathrm{Irr}}
\def\Rad{\mathrm{Rad}}
\def\Res{\mathrm{Res}}
\def\Resf{\mathrm{Resf}}
\def\tr{\mathrm{tr}}
\def\height{\mathrm{ht}}
\def\Tr{\mathrm{Tr}}
\def\wt{\mathrm{wt}}


%%%%% Diagrams %%%%% 
\usepackage{etex} %fixes the fight that pictex has with every other drawing package
\usepackage{pictexwd}
%\input xy
%\xyoption{all}
\usepackage{tikz}
	\usepgflibrary[patterns] % ConTEXt and pure pgf 
	\usetikzlibrary{patterns} % LATEX and plain TEX when using TikZ 
	\usepgflibrary{shapes.geometric}
        \usetikzlibrary{arrows, calc, positioning}

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

\def\ShiftX{\pgftransformxshift}
\def\ShiftY{\pgftransformyshift}

%PARTITIONS
\newcounter{r}
\newcounter{s}




%Shortcut for a partition, to be used in TikZ environment.
%Example: \begin{tikzpicture} \Part{5,4,2} \end{tikzpicture}
\newcommand\Part[1]{
        \setcounter{r}{1}
	 \foreach \x in {#1}{
 	{\ifnum\value{r}=1
		\draw (0,\value{r}-1)--(\x,\value{r}-1); 
		\fi}
	\draw (0,\value{r}) to (\x,\value{r});
   	\foreach \y in {0, ..., \x} {\draw (\y,\value{r})--(\y,\value{r}-1);}
	\addtocounter{r}{1}
 }}
 \def\PartUNIT{.175}
%Self-contained tikz images for \Part above, to be used in math mode.
%Example: $\PART{5,4,2}$ or $\sPART{5,4,2}$
\newcommand{\PART}[1]{%Good for displayed, moderate sized partitions.
\begin{matrix}
\begin{tikzpicture}[scale=.35, yscale=-1] 
	\Part{#1}\node at (.5,0){};
\end{tikzpicture}
\end{matrix}
}
\newcommand{\sPART}[1]{%Good for subscripts.
\begin{matrix}
\begin{tikzpicture}[scale=.175, yscale=-1] 
	\Part{#1}\node at (.5,0){};
\end{tikzpicture}
\end{matrix}
}

%Outline of a partition. Use in tikzpicture environment. 
%Example: \begin{tikzpicture} \EmptyPart{5,4,2} \end{tikzpicture}
\newcommand\EmptyPart[1]{
        \setcounter{r}{0}
        \setcounter{s}{0}
	 \foreach \x in {#1}{
	 \draw (\value{s}, \value{r}) to (\x, \value{r}) to (\x, \value{r}+1);
%   	\foreach \y in {0, ..., \x} {\draw (\y,\value{r})--(\y,\value{r}-1);}
	\addtocounter{r}{1}
        \setcounter{s}{\x}
 }
 \draw (\value{s}, \value{r}) to (0, \value{r}) to (0,0);
 }
 %Stand-alone code for \EmptyPart. Use in math environment.
 %Example: $\ePART{5,4,2}$
 \newcommand{\ePART}[1]{
\begin{matrix}
\begin{tikzpicture}[scale=.35, yscale=-1] 
	\EmptyPart{#1}\node at (.5,0){};
\end{tikzpicture}
\end{matrix}
}
 
%Shortcut for a tableau, to be used in TikZ environment.
%Example: \begin{tikzpicture} \Tableau{{1,1,1,2,3},{2,2,4,4},{3,4}} \end{tikzpicture}
\newcommand\Tableau[1]{
        \foreach \x [count = \c from 1] in {#1} {
		\foreach \y [count = \d from 1] in \x{
			\node at (\d-.5,\c-.5) {\scriptsize$\y$}; %Want to change the color of the numbers? insert \color{???} just after \scriptsize
			\draw (\d,\c) to (\d,\c-1);
			{\ifnum\d=1
				\draw (0,\c) to (0,\c-1);
				\fi}
			\setcounter{r}{\d}
		}
		{\ifnum\c=1
			\draw (0,0)--(\value{r},0);
			\fi}
		\draw(0,\c) to (\value{r},\c);
		\setcounter{s}{\c}}}
		
 %Stand-alone code for \Tableau. Use in math environment.
 %Best for displayed environments.
 %Example: $\TBL{{1,1,1,2,3},{2,2,4,4},{3,4}}$
\newcommand{\TBL}[1]{
\begin{matrix}
\begin{tikzpicture}[scale=.35, yscale=-1] 
	\Tableau{#1}\node at (.5,0){};
\end{tikzpicture}
\end{matrix}
}
%Shortcut for a small tableau, to be used in TikZ environment.
% Best for in-line or subscripts.
%Example: \begin{tikzpicture} \sTableau{{1,1,1,2,3},{2,2,4,4},{3,4}} \end{tikzpicture}
\newcommand\sTableau[1]{
        \foreach \x [count = \c from 1] in {#1} {
		\foreach \y [count = \d from 1] in \x{
			\node at (\d-.5,\c-.5) {\tiny$\y$}; %Want to change the color of the numbers? insert \color{???} just after \scriptsize
			\draw (\d,\c) to (\d,\c-1);
			{\ifnum\d=1
				\draw (0,\c) to (0,\c-1);
				\fi}
			\setcounter{r}{\d}
		}
		{\ifnum\c=1
			\draw (0,0)--(\value{r},0);
			\fi}
		\draw(0,\c) to (\value{r},\c);
		\setcounter{s}{\c}}}
%Stand-alone code for \sTableau. Use in math environment.
%Best for in-line or subscripts.
%Example: $\sTBL{{1,1,1,2,3},{2,2,4,4},{3,4}}$		
\newcommand{\sTBL}[1]{
\begin{matrix}
\begin{tikzpicture}[scale=.25, yscale=-1] 
	\sTableau{#1}\node at (.5,0){};
\end{tikzpicture}
\end{matrix}
}



\newcommand{\PartB}[1]{
 \foreach \x [count=\s from 1] in {#1}{
 	{\ifnum\s=1
		\draw (0,\s-1)--(\x,\s-1); 
		\fi}
   \draw (0,\s) to (\x,\s);
   \foreach \y in {0, ..., \x} {\draw (\y,\s)--(\y,\s-1);}
 }}

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

%Self-contained tikz images for \Part and \fPart above. 
\newcommand{\PARTB}[1]{
\begin{tikzpicture}[xscale=\UNITB, yscale=-\UNITB] 
	\Part{#1}
\end{tikzpicture}
}
\newcommand{\fPARTB}[2]{
\begin{tikzpicture}[xscale=\UNITB, yscale=-\UNITB]
	\fPart{#1}{#2}
\end{tikzpicture}
}
%This is the lattice behind all the examples below. 
\def\Lattice{
%Sets the coordinates for the partitions of 0 to 5, grouped by level. Adjust y-coordinates to spread out levels piecemeal, or just adjust overall yscale in the TikZ environment.
	\coordinate (0) at (0,.4); %level 0
	\coordinate (11) at (0,1.2); %level 1
	\coordinate (21) at (-1,2);\coordinate (22) at (1,2); %level 2
	\foreach \x in {1, ..., 3}{\coordinate (3\x) at (-4+2*\x,3);} %level 3
	\foreach \x in {1, ..., 5}{\coordinate (4\x) at (-6+2*\x,4.5);} %level 4
	\foreach \x in {1, ..., 7}{\coordinate (5\x) at (-8+2*\x,6);} %level 5
%edges in lattice:
	%level 0 -> 1
	\draw (0) to node[left] {$a$} (11); 
	%level 1 -> 2
	\draw (11) to node[above left] {$b$} (21) 
		(11) to node[above right] {$c$} (22);
	\draw (21) to node[above left] {$e$} (31) 
	%level 2 -> 3
		(21) to node[below left] {$f$} (32) 
		(22) to node[above left] {$g$} (32) 
		(22) to node[above right] {$h$} (33);
	%level 3 -> 4
	\draw (31) to node[above left] {$i$}  (41) 
		(31) to node[left] {$j$}  (42) 
		(32) to node[above left] {$k$}  (42) 
		(32) to node[left] {$l$}  (43) 
		(32) to node[above right] {$m$}  (44) 
		(33) to node[right] {$n$}  (44) 
		(33) to node[above right] {$o$}  (45);
	%level 4 -> 5
	\draw (41) to node[left] {$p$}  (51) 
		(41) to node[left] {$q$}  (52) 
		(42) to node[left] {$r$}  (52) 
		(42) to node[left] {$s$}  (53) 
		(42) to[bend left=5] node[pos=.35, above right] {$t$}  (54) 
		(43) to[bend right=5] node[pos=.35, below right] {$u$}  (53)
		(43) to[bend left=5] node[pos=.35, above right] {$v$}  (55) 
		(44) to[bend right=5] node[pos=.35, below right] {$w$}  (54) 
		(44) to[bend right=5] node[right] {$x$}  (55) 
		(44) to node[above right] {$y$}  (56) 
		(45) to node[right] {$z$}  (56) 
		(45) to node[above right] {$?$}  (57);
%partitions in lattice:
\begin{scope}[every node/.style={fill=white}]
	%level 0
	\node at (0) {$\emptyset$};
	%level 1
	\node at (11) {\PARTB{1}};
	%level 2
	\node at (22) {\PARTB{1,1}};
	\node at (21) {\PARTB{2}};
	%level 3
	\node at (33) {\PARTB{1,1,1}};
	\node at (32) {\PARTB{2,1}};
	\node at (31) {\PARTB{3}};
	%level 4
	\node at (45) {\PARTB{1,1,1,1}};
	\node at (44) {\PARTB{2,1,1}};
	\node at (43) {\PARTB{2,2}};
	\node at (42) {\PARTB{3,1}};
	\node at (41) {\PARTB{4}};
	%level 5
	\node at (57) {\PARTB{1,1,1,1,1}};
	\node at (56) {\PARTB{2,1,1,1}};
	\node at (55) {\PARTB{2,2,1}};
	\node at (54) {\PARTB{3,1,1}};
	\node at (53) {\PARTB{3,2}};
	\node at (52) {\PARTB{4,1}};
	\node at (51) {\PARTB{5}};
\end{scope}
\foreach \x in {-4,0,4} {\node at (\x , 6.75) {$\vdots$}; }
}

%%%%%% Diagrams %%%%% 
%\usepackage{etex} %fixes the fight that pictex has with every other drawing package
%\usepackage{pictexwd}
%\usepackage{tikz}
%\usepgflibrary{shapes.geometric}
%\usetikzlibrary{arrows, calc, positioning}
%%\usetikzlibrary{arrows}
%\def\ShiftX{\pgftransformxshift}
%\def\ShiftY{\pgftransformyshift}

\tikzstyle{V}=[draw, fill =black, circle, inner sep=0pt, minimum size=1.5pt]
\tikzstyle{wV}=[draw, fill =white, circle, inner sep=0pt, minimum size=4.5pt]
\tikzstyle{bV}=[draw, fill =black, circle, inner sep=0pt, minimum size=4.5pt]
\tikzstyle{over}=[draw=white,double=black,line width=2pt, double distance=.5pt]


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


%\def\Ez[#1]{\draw [over, bend left=75] (1,#1+1) to (1-.4,#1+.7)  (1-.4,#1+.3)  to (1,#1) ;
%		\draw[densely dotted]  (1-.4,#1)--(1-.4,#1+1) ; \node[V] at (1-.4,#1+.3){}; \node[V] at  (1-.4,#1+.7){};}
%\def\Ek[#1][#2]{\draw [over, bend right=75] (#2,#1+1) to (#2+.4,#1+.7)  (#2+.4,#1+.3)  to (#2,#1) ;
%		\draw[densely dotted]  (#2+.4,#1)--(#2+.4,#1+1) ; \node[V] at (#2+.4,#1+.3){}; \node[V] at  (#2+.4,#1+.7){};}

\def\Tops[#1][#2][#3]{%1=pole locations, 2=top, 3=k 
	\foreach\x in {#1}{
		\draw (#2,\x+.15) -- (#2+.1, \x+.15) (#2, \x-.15) -- (#2+.1, \x-.15) ;
		\draw (#2+.1,\x) arc (0:360:.75mm and 1.5mm);}
	%Nodes 
	\foreach \x in {1,...,#3} {\draw (#2,\x)  to (#2+.05,\x); \node[V] at (#2+.05,\x){};}
	}
\def\Bottoms[#1][#2][#3]{%1=pole locations, 2 = bottom, 3=top 
	\foreach\x in {#1}{
		\draw (#2, \x+.15) -- (#2-.1, \x+.15) (#2, \x-.15) -- (#2-.1, \x-.15) ;
		\draw (#2-.1, \x+.15) arc (90:270:.75mm and 1.5mm);}
	%Nodes 
	\foreach \x in {1,...,#3} {\draw (#2, \x)  to (#2-.05, \x); \node[V] at (#2-.05, \x){};}
	}
\def\Caps[#1][#2,#3][#4]{%1=pole locations, 2 = bottom, 3=top, 4=k 
	\Tops[#1][#3][#4]
	\Bottoms[#1][#2][#4]
	}
\def\Pole[#1][#2,#3]{%1=horizontal location, 2 = bottom, 3=top
	\shade[left color=white,right color=white] (#2,#1+.15) rectangle (#3,#1-.15);
	\draw[over] (#2,#1+.15) to (#3,#1+.15) (#2,#1-.15) to (#3,#1-.15) ;}
\def\Label[#1,#2][#3][#4]{%1,2 = top/bot position, 3=i, 4=label
	\node[right] at (#2+.1,#3) {#4};
	\node[left] at (#1-.1,#3) {#4};		}
\def\Nodes[#1][#2]{
	 \foreach \x in {1,...,#2} {\node[V] at (#1,\x){};	}
	}
\def\PoleCaps[#1][#2,#3]{%1=pole location, 2 = bottom, 3=top, 4=k 
	\foreach\x in {#1}{
		\draw (#2,\x+.15) -- (#2-.1,\x+.15) (#2,\x-.15) -- (#2-.1,\x-.15) ;
		\draw (#2-.1,\x+.15) arc (0:-180:1.5mm and .75mm);}
	\foreach\x in {#1}{
		\draw (#3,\x+.15) -- (#3+.1,\x+.15) (#3,\x-.15) -- (#3+.1,\x-.15) ;
		\draw (#3+.1,\x+.15) arc (0:360:1.5mm and .75mm);}
	}
\def\PoleTwist[#1,#2]{%1 = bottom, 2=top
	\foreach \x/\y in {-1/1L, -.7/1R, 0/2L, .3/2R}{\coordinate(T\y) at (#2,\x); \coordinate(B\y) at (#1,\x);}
	\draw[thin] (B1R) .. controls ++(#2*.5-#1*.5-.1,0) and ++(-#2*.5+#1*.5-.1,0) ..  (T2R)
			(B1L)   .. controls ++(#2*.5-#1*.5+.1,0) and ++(-#2*.5+#1*.5+.1,0) ..    (T2L) ;
	\draw[line width=2pt, white]
			(#1,.15)  .. controls +(#2*.5-#1*.5,0) and +(-#2*.5+#1*.5,0) ..   (#2,-.85) ;
	\draw[thin,over] 
		(B2R) .. controls ++(#2*.5-#1*.5+.1,0) and ++(-#2*.5+#1*.5+.1,0) ..  (T1R) 
			(B2L)  .. controls +(#2*.5-#1*.5-.1,0) and +(-#2*.5+#1*.5-.1,0) ..   (T1L) ;
			}


\def\SymPolesCaps[#1,#2][#3]{%1 = Vertical position, 2=k
	\draw (#1,.3) -- (#1-.1,.3) (#1,.15) -- (#1-.1, .15) ;
	\draw (#1-.1, .3) arc (0:-180:2pt and 1.5pt);
	\draw (#1,#3+.7) -- (#1-.1,#3+.7) (#1,#3+.85) -- (#1-.1,#3+.85) ;
	\draw (#1-.1,#3+.85)  arc (0:-180:2pt and 1.5pt);
	\draw (#2,.3) -- (#2+.1, .3) (#2, .15) -- (#2+.1, .15) ;
	\draw (#2+.1, .3) arc (0:360:2pt and 1.5pt);
	\draw (#2, #3+.7) -- (#2+.1, #3+.7) (#2, #3+.85) -- (#2+.1, #3+.85) ;
	\draw (#2+.1, #3+.85) arc (0:360:2pt and 1.5pt);}


%************Commutative diagrams**********************

\def\mapright#1{\smash{\mathop
        {\longrightarrow}\limits^{#1}}}

\def\mapleftright#1{\smash{\mathop
        {\longleftrightarrow}\limits^{#1}}}


\def\mapsrightto#1{\smash{\mathop
        {\longmapsto}\limits^{#1}}}

\def\mapleft#1{\smash{
   \mathop{\longleftarrow}\limits^{#1}}}

\def\mapdown#1{\Big\downarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}

\def\lmapdown#1{{\hbox{$\scriptstyle#1$}}
\llap {$\vcenter{\hbox{\Big\downarrow}}$} }

\def\mapupdown#1{\Big\updownarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}

\def\lmapupdown#1{{\hbox{$\scriptstyle#1$}}
\llap {$\vcenter{\hbox{\Big\updownarrow}}$} }

\def\mapup#1{\Big\uparrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapne#1{\Big\nearrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapse#1{
%{$\vcenter{
\hbox{$\scriptstyle#1$}
%$}
\rlap{ $\vcenter{\hbox{$\searrow$}}$ }  }
\def\mapnw#1{\Big\nwarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapsw#1{
%\Big
\swarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}


\newcommand{\posleq}[1]{
	\hspace{0.1cm}
	\begin{tikzpicture}
	\draw (-0.8ex, -0.5ex) -- (0.8ex, -0.5ex);
	\draw (-0.8ex, 0.4ex) -- (0.7ex, -0.2ex);
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%\newcommand{\yaping}[1]{\textcolor{purple}{$[$ Yaping: #1 $]$}}

\makeatletter
\renewcommand{\@makefnmark}{\mbox{\textsuperscript{}}}
\makeatother


\title{Title}
\author{
Author \\
authoremail@email.address.adm
%Department of Mathematics and Statistics \\
%University of Melbourne \\
%Parkville VIC 3010 Australia \\
% \\
\\
}
\date{\today}

\lhead{short title, Author version: \today}
\rhead{}
\usetikzlibrary{arrows.meta}

\begin{document}

\maketitle

%\vspace{-2em}
%\begin{center}
%{\sl Dedicated to Peter Littelmann}
%Littelmann}
%\end{center}


\begin{abstract}
\noindent
This is a TeX template provided to students in Advanced Discrete Math MAST90030 University of Melbourne 
by Arun Ram July 2025.
\end{abstract}

\keywords{symmetric functions, crystals, binomial theorems, and hypergeometric functions}
\footnote{AMS Subject Classifications: Primary 05E05; Secondary  20G99.}

\setcounter{section}{-1}
%\setcounter{tocdepth}{4}
\tableofcontents





\newpage

\section{Introduction}

\subsection{Nice story}

Nice helpful overview story goes here.


\section{Main content}

In the culture of academic science,
any document that does not have accurate and careful referencing and a carefully produced bibiliography 
and attentive and conscientious inline intext chapter-verse referencing will usually receive negative marks.
Sometimes documents without suitable referencing are referred to by the collective term `plaigiarism'.





\newpage


\begin{thebibliography}{20}

\bibitem[Ai79]{Ai79}
M.\ Aigner, \textsl{Combinatorial Theory}, Springer-Verlag (1979).

\bibitem[Be71]{Be71}
C.\ Berge, \textsl{Principles of Combinatorics}, Academic Press (1971).

\bibitem[Ch78]{Ch78}
D.I.A.\ Cohen, \textsl{Basic Techniques of Combinatorial Theory}, Wiley (1978).

\bibitem[Co74]{Co74}
L.\ Comtet, \textsl{Advanced Combinatorics}, D.\ Reidel (1974).

\bibitem[Fe57]{Fe57}
W.\ Feller, \textsl{An Introduction to Probability Theory and its Applications}, Volume 1 2nd ed.\ John-Wiley (1957).

\bibitem[Go72]{Go72}
H.W.\ Gould, \textsl{Combinatorial Identities -- A standardized table of 500 binomial coefficient summations}, Gould (1972).

\bibitem[GKP89]{GKP89}
R.L.\ Graham, D.E.\ Knuth, O.\ Patashnik, \textsl{Concrete Mathematics - A Foundation for Computer Science}, Addison-Wesley (1989).

\bibitem[Kn68]{Kn68}
D.E.\ Knuth, \textsl{The Art of Computer Programming, Volume I: Fundamental Algorithms}, 2nd ed.\ Addison-Wesley (1968).

\bibitem[Li68]{Li68}
C.L.\ Liu, \textsl{Introduction to Combinatorial Mathematics}, McGraw-Hill (1968).

\bibitem[Mac]{Mac} I.G.\ Macdonald, 
\textsl{Symmetric functions and Hall polynomials}, 
Second edition, Oxford Mathematical Monographs, Oxford University Press, New York, 1995. 
ISBN: 0-19-853489-2, MR1354144.

\bibitem[Ri58]{Ri58}
J.\ Riordan, \textsl{An Introduction to Combinatorial Analysis}, Wiley (1958).

\bibitem[RM78]{Rm78}
S.\ Roman, \textsl{The Umbral Calculus}, Academic Press (1978).

\bibitem[Ro75]{Ro75}
G.-C.\ Rota, \textsl{Finite Operator Calculus}, Academic Press (1975).

\bibitem[St86]{St86}
R.P.\ Stanley, \textsl{Enumerative Combiantorics Volume I}, Wadsworth \& Brooks/Cole Mathematics Series (1986).

\bibitem[To85]{To73}
I.\ Tomescu, \textsl{Introduction to Combinatorics}, Coliet's (1973).

\bibitem[To85]{To85}
I.\ Tomescu, \textsl{Problems in Combinatorics and Graph Theory}, Wiley-Interscience (1985).

\bibitem[Wh01]{Wh01}
W.A.\ Whitworth, \textsl{Choice and Chance}, Bell (1901).






\end{thebibliography}




\end{document}



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