

\newcommand{\bx}{\mathbf{x}}
\newcommand{\schur}[1][]{s_{#1}(\bx)}
\newcommand{\pn}{\mathcal{P}[n]}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\cx}{\complex[\bx]}
\newcommand{\integers}{\mathbb{Z}}
\newcommand{\zplus}{\integers_+}
\newcommand{\lie}{\mathfrak}
\newcommand{\wnot}{w_0}
\newcommand{\lr}{c_{\lambda\mu}^{\,\nu}}
\newcommand{\lrw}[1][w]{c_{\lambda\mu}^{\,\nu}(#1)}
\newcommand{\lrwk}[1][w]{c_{k\lambda,k\mu}^{\,k\nu}(#1)}
\newcommand{\supstd}[1][]{T^{\circ}_{#1}}
\newcommand{\ssword}[1][]{b^{\circ}_{#1}}
\newcommand{\zero}{\mathbf{0}}
\newcommand{\mub}{\overline{\mu}}
\newcommand{\zetab}{\overline{\zeta}}
\newcommand{\checkthis}[1][Check this]{\textcolor{red}{#1}}
\newcommand{\dchar}[1][w,\mu]{\chi_{#1}(\bx)}
\newcommand{\bighive}{\mathbf{\Delta}}
\newcommand{\ps}[1][]{\overline{#1}}
\newcommand{\lmn}{\lambda, \mu, \nu}
\newcommand{\sd}{\partial}
\newcommand{\bsd}{{\boldsymbol \partial}}
\renewcommand{\iff}{\Leftrightarrow}
\newcommand{\kcone}{\koganH(\text{ --} \,, \wnot w)}
\newcommand{\kzcone}{\koganHz(\text{ --} \,, \wnot w)}
\newcommand{\rootthree}{1.7320508}

% \newcommand{}{}
% \newcommand{}{}
% \newcommand{}{}
% \newcommand{}{}
% \newcommand{}{}
% \newcommand{}{}

\DeclareMathOperator{\demcrys}{Dem}
\DeclareMathOperator{\dem}{T}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\U}{\mathfrak{U}}
\DeclareMathOperator{\ch}{char}
\DeclareMathOperator{\Tab}{Tab}
\DeclareMathOperator{\GT}{GT}
\DeclareMathOperator{\GTz}{GT_\integers}
\DeclareMathOperator{\kogan}{K}
\DeclareMathOperator{\koganz}{K_\integers}
\DeclareMathOperator{\koganH}{\kogan^{\text{\textup{\footnotesize{}Hive}}}}
\DeclareMathOperator{\koganHz}{\kogan_\integers^{\text{\textup{\footnotesize{}Hive}}}}
\DeclareMathOperator{\Hive}{Hive}
\DeclareMathOperator{\Hivez}{\Hive_\integers}
\DeclareMathOperator{\bg}{\mathbf{\Gamma}}
\DeclareMathOperator{\len}{len}
%\DeclareMathOperator{\sd}{\partial}
%\DeclareMathOperator{\bsd}{{\boldsymbol \partial}}
\DeclareMathOperator{\cont}{content}
\DeclareMathOperator{\horn}{Horn}
% \DeclareMathOperator{}{}
% \DeclareMathOperator{}{}
% \DeclareMathOperator{}{}
% \DeclareMathOperator{}{}
% \DeclareMathOperator{}{}
