%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{ \h }{ \mathfrak{h} }   %%%% Cartan subalgebra %%%%
\newcommand{ \g }{ \mathfrak{g} }   %%%% Lie algebra %%%%
\newcommand{ \rroot }{ \Phi }       %%%% real root system %%%%

\newcommand{ \iw }{ P }             %%%% the set of integral weights %%%%
\newcommand{ \pdiw }{ P_{\geq -1} } %%%% the set of pre-dominant integral weights %%%%
\newcommand{\sigiw}{ P_{\!\mathrm{sig}} }
\newcommand{\nulliw}{P_0}
\newcommand{ \Gpdiw }[1]{ \pdiw^{\alpha_{#1}} }
\newcommand{\diw}{ \iw_{\geq 0} }
\newcommand{\infpdiw}{ \pdiw^{\rm \,inf} }
\newcommand{\finpdiw}{ \pdiw^{\rm \,fin} }
\newcommand{ \nliw}{ P_{0} }







\newcommand{ \G }[2]{ \mathcal{G}_{#1}( #2 ) }
\newcommand{ \F }[3]{ \mathcal{F}^{(#1)}_{#2}( #3 ) }
\newcommand{ \GF }[2]{ \mathcal{G}_{\alpha_{#1}}( #2 ) }
\newcommand{ \US }[3]{ \mathcal{F}^{(#1)}_{\alpha_{#2}}( #3 ) }




\newcommand{ \diag }[1]{ \mathrm{D}( #1 ) }
\newcommand{ \twdiag }[2]{ \mathrm{D}^{ \alpha_{ #1 } }( #2 ) }
\newcommand{ \gendiag }[2]{ \mathrm{GD}^{ \alpha_{ #1 } }( #2 ) }
\newcommand{ \Gdiag }[2]{ \mathrm{GD}^{ \alpha_{ #1 } }( #2 ) }



\newcommand{\Hook}[2]{ \mathrm{H}_{#1} \left( #2 \right)}
\newcommand{\inv}[1]{ \Phi \left( #1 \right) }
\newcommand{\htt}[1]{ \mathrm{ht} \left( #1 \right)}



\newcommand{ \Path }[1]{ \mathrm{Path}( #1 ) }
\newcommand{ \basPath }[2]{ \mathrm{Path}_{ \alpha_{#1} }( #2 ) }
\newcommand{ \twPath }[2]{ s_{#1} \left( \mathrm{Path}( s_{ \alpha_{#1} }( #2 ) ) \right) }
\newcommand{ \GPath }[2]{ \mathrm{GPath}_{ \alpha_{#1} }( #2 ) }
\newcommand{\QPath}[2]{\mathrm{QPath}_{\alpha_{#1}}\!\left( #2 \right)}


\newcommand{ \shiftable }[1]{ #1 \mbox{-shiftable} }

%\newcommand{ \Path }[1]{ \mathrm{Path}( #1 ) }
\newcommand{ \MPath }[1]{ \mathrm{MPath}( #1 ) }
\newcommand{ \STab }[1]{ \mathrm{STab} ( #1 ) }
\newcommand{ \Red }[1]{ \mathrm{Red} ( #1 ) }





\newcommand{\shape}[1]{ {\diag{#1}^{\!\vee}} }

\newcommand{\covered}{ \stackrel{\,_{\bullet}}{<} }
\newcommand{\bruhat}[3]{ #1 \stackrel{ #2 }{ \triangleleft } #3 }
\newcommand{\horder}[1]{ \stackrel{ #1 }{ \triangleleft } }

\newcommand{\PSTab}[1]{ \mathrm{PSTab}( #1 ) }
\newcommand{\Numb} [1]{\mathrm{Numb}( #1 )}
\newcommand{\MNumb}[1]{\mathrm{MNumb}( #1 )}

\newcommand{\SLab} [1]{\mathrm{SLab}( #1 )}




\newcommand{\reach}[2]{ #2 \mbox{ is reachable from } #1 }


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%  q-hook formula   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\newcommand{\qq}{ \mathbf{q} }
%\renewcommand{\erat}[1]{ \frac{ \qq^{#1} }{ 1 - \qq^{#1} } }
\newcommand{\erat}[1]{ \frac{ e^{#1} }{ 1 - e^{#1} } }
\newcommand{\qrat}[1]{ \frac{ q^{\htt{#1}} }{ 1 - q^{\htt{#1}} } }

\newcommand{\qqrat}[1]{ \frac{ \qq^{#1} }{ 1 \!-\! \qq^{#1} }     }

\newcommand{\terat}[1]{ \textstyle{ \erat{#1} } }
\newcommand{\txqqrat}[1]{ \textstyle{ \qqrat{#1} } }



\newcommand{\Skew}{ \mathrm{\Sigma} }
\newcommand{\SSkew}{ \mathrm{\Sigma\!\!\!\;\Sigma} }
\newcommand{\strict}{ {}^{\times} }

%%%% W-ordered set %%%%

\newcommand{\Jf}[1]{ \mathrm{J_f}\left( #1 \right) }
\newcommand{\Part}[1]{ \mathsf{A}\left( #1 \right) }
\newcommand{\Chain}[1]{ \mathrm{Chain}\left( #1 \right) }

\newcommand{\xx}{\mathbf{x}}
\newcommand{\xrat}[1]{ \frac{ \xx^{#1} }{ 1-\xx^{#1} } }
\newcommand{\xxrat}[1]{ \frac{ \xx^{#1} }{ 1-\xx^{#1} } }
\newcommand{\LE}[1]{ \mathcal{L}\left( #1 \right) }



%%%%%%%%%%%%%%%%%



%\newcommand{\Ref}[1]{ \mathrm{Ref}\left( #1 \right) }


\newcommand{\ACmove}[2]{ \langle #1 ,\, #2 ^\vee \rangle \!=\! \langle #2 ,\, #1 ^\vee \rangle \!=\! 2 }
\newcommand{\Shift}[1]{ #1 \mbox{-shiftable} }
\newcommand{\shift}[2]{ #1 \mbox{ is } #2 \mbox{-shiftable} }
\newcommand{\notshift}[2]{ #1 \mbox{ is not } #2 \mbox{-shiftable} }
\newcommand{\comm}[3]{ #1 \mbox{ is } #2 \mbox{-commutative at } #3}
\newcommand{\notcomm}[3]{ #1 \mbox{ is not } #2 \mbox{-commutative at } #3}

\newcommand{\Rpp}[1]{ \mathrm{Rpp} \left( #1 \right) }


\newcommand{\chose}[2]{ \left( \!\! \begin{array}{c} {\small #1} \\ {\small #2} \end{array} \!\! \right) }
\newcommand{\sigset}{ \{ 1, 0 ,-1 \} }
\newcommand{\parity}{\epsilon}

\newcommand{\position}[2]{ \stackrel{#1}{\stackrel{\smile}{#2}} }
\newcommand{\oo}[1]{ \stackrel{ {}_{{}_{{}_{{}_{\circ}}}} }{#1} }

%\newcommand{\return}{ \!\!\! \\ \!\!\! }













\newcommand{\MSSkew}{ \mathrm{M}\SSkew }
\newcommand{\Comm}[1]{ \mbox{\( #1 \)-commutative} }
\newcommand{\CQmove}[3]{
 #1 \isnot \Shift{\alpha_{#3}}, #1 \neq #2 \alpha_{#3} \et \langle #1 ,\, \alpha_{#3}^\vee \rangle = #2 2
}
\newcommand{\Stair}[3]{ \mathcal{S}^{( #1 )}_{\alpha_{#2}}\left( #3 \right) }
\newcommand{\RSTab}[1]{ \mathrm{RSTab}\left( #1 \right)}

\newcommand{\SPath}[1]{ \mathrm{SPath}{ \left( #1 \right) } }
\newcommand{\SShape}[1]{ \mathrm{SShape}{ \left( #1 \right) } }
\newcommand{\Init}[1]{ \mathrm{Init} \left( #1 \right) }

\newcommand{\corder}{\prec}
\newcommand{\aal}{ {\boldsymbol \alpha} }
\newcommand{\shook}[3]{ \mathrm{M}^{#1}_{#2}\! \left( #3 \right) }
\newcommand{\dd}[1]{ {}_{        {}\atop{ #1 }       } }

\newcommand{\Q}[1]{ \mathrm{Q}^{\alpha_{#1}}   }
\newcommand{\bas}[1]{ \mathrm{bas}^{\alpha_{#1}}   }
\newcommand{\tw}[1]{ \mathrm{tw}^{\alpha_{#1}}   }

\newcommand{\parset}{ \{ 1 , -1 \} }


\newcommand{\heap}{\preceq}
\newcommand{\norder}{\stackrel{0}{<}}
\newcommand{\incomp}{\,||\,}
