% Packages
%\usepackage{amsmath}              % provides \underset, etc.
%\usepackage{amssymb}              % provides more symbols like semidirect product
%\usepackage{amsopn}                % provides \DeclareMathOperator, \mathbb, etc.
%\usepackage{enumerate}            % more powerful enumerate environments
%usepackage{mathtools}            % \coloneqq
%\usepackage[margin=1in]{geometry} % margins only
%\usepackage{microtype}            % nicer text spacing
%\usepackage{thmtools}             % interface to amsthm
%\usepackage{cleveref}
%\usepackage{url}
\usepackage{subcaption}    % subfigure environment
\usepackage{cancel}
%\usepackage{array}
\usepackage{xcolor}           % \color
\usepackage{booktabs} % \toprule etc.

% For use with tabu
%\tabulinesep=1.2mm

% For use with subcaption
\newdimen\figrasterwd
\figrasterwd\textwidth

% Math operators
\DeclareMathOperator{\aft}{aft}
\DeclareMathOperator{\arm}{arm}
\DeclareMathOperator{\baj}{baj}
\DeclareMathOperator{\Ch}{ch}
\DeclareMathOperator{\comaj}{comaj}
\DeclareMathOperator{\des}{des}
\DeclareMathOperator{\Des}{Des}
\DeclareMathOperator{\Fl}{Fl}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Hilb}{Hilb}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\inv}{inv}
\DeclareMathOperator{\Inv}{Inv}
\DeclareMathOperator{\leg}{leg}
\DeclareMathOperator{\maj}{maj}
\DeclareMathOperator{\median}{median}
\DeclareMathOperator{\PP}{PP}
\let\Re\ralax\DeclareMathOperator{\Re}{Re}
\DeclareMathOperator{\RSYT}{RSYT}
\DeclareMathOperator{\sinc}{sinc}
\DeclareMathOperator{\size}{size}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\SSYT}{SSYT}
\DeclareMathOperator{\stat}{stat}
\DeclareMathOperator{\SYT}{SYT}
\DeclareMathOperator{\uni}{uni}
\DeclareMathOperator{\lcc}{lcc}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\W}{W}
\newcommand{\gmdn}[1]{{g^{\{\underline{#1}\}^d} }}
\newcommand{\bbinom}{\genfrac{[}{]}{0pt}{}}

\newcommand{\red}[1]{{\color{red} #1}}
\newcommand{\green}[1]{{\color{green} #1}}

\newcommand{\rank}{b}  %% changed to respect the liturature

% Convenient shortcuts
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bk}{\mathbbm{k}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bQ}{\mathbb{Q}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cIH}{\mathcal{IH}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cZ}{\mathcal{Z}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\fh}{\mathfrak{h}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fn}{\mathfrak{n}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}}
\newcommand{\fsl}{\mathfrak{sl}}
\newcommand{\fS}{\mathfrak{S}}
\newcommand{\fU}{\mathfrak{U}}

% Convenient general macros
\newcommand{\too}[1]{\stackrel{#1}{\to}}
\newcommand{\mbinom}[2]{%
  \left.\mathchoice%
    {\left(\kern-0.48em\binom{#1}{#2}\kern-0.48em\right)}
    {\big(\kern-0.30em\binom{\smash{#1}}{\smash{#2}}\kern-0.30em\big)}
    {\left(\kern-0.30em\binom{\smash{#1}}{\smash{#2}}\kern-0.30em\right)}
    {\left(\kern-0.30em\binom{\smash{#1}}{\smash{#2}}\kern-0.30em\right)}
  \right.
}

% Mark Haiman's tableaux macro, modified to put dashed lines around empty boxes
\setlength{\unitlength}{0.06em}
\newlength{\cellsize} \setlength{\cellsize}{18\unitlength}
\newsavebox{\cell}
\sbox{\cell}{\begin{picture}(18,18)
\put(0,0){\line(1,0){18}}
\put(0,0){\line(0,1){18}}
\put(18,0){\line(0,1){18}}
\put(0,18){\line(1,0){18}}
\end{picture}}
\newsavebox{\dashedcell}
\sbox{\dashedcell}{\begin{picture}(18,18)
\multiput(0,0)(5,0){4}{\line(1,0){3}}
\multiput(0,0)(0,5){4}{\line(0,1){3}}
\multiput(18,0)(0,5){4}{\line(0,1){3}}
\multiput(0,18)(5,0){4}{\line(1,0){3}}
\end{picture}}
\newcommand\cellify[1]{\def\thearg{#1}\def\nothing{}%
\ifx\thearg\nothing
\hbox to 0pt{ \hss}\else%
\hbox to 0pt{\usebox{\cell} \hss}\fi%
\vbox to \cellsize{
\vss
\hbox to \cellsize{\hss$#1$\hss}
\vss}}
\newcommand\tableau[1]{\vtop{\let\\\cr
\baselineskip -16000pt \lineskiplimit 16000pt \lineskip 0pt
\ialign{&\cellify{##}\cr#1\crcr}}}


\theoremstyle{plain}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Fact}[Theorem]{Fact}
\newtheorem{Conjecture}[Theorem]{Conjecture}
\newtheorem{Question}[Theorem]{Question}
\newtheorem{Problem}[Theorem]{Open Problem}
\newtheorem{Summary}[Theorem]{Summary}

\theoremstyle{definition}
\newtheorem{Notation}[Theorem]{Notation}
%\declaretheorem[style=definition,sibling=Theorem]{Definition}
\newtheorem{Definition}[Theorem]{Definition}
\newtheorem{Example}[Theorem]{Example}
\newtheorem{Remark}[Theorem]{Remark}
\newtheorem{Aside}[Theorem]{Aside}