% Linear Algebra

%%%% - vector

\newcommand{\vect}[1]{ \mathbf{#1} }

\newcommand{\VEC}[1]{ {\left[ \!\!\begin{array}{c} #1 \end{array}\!\! \right]} } % number vector as point
\newcommand{\COORD}[1]{ {\left[ \!\!\begin{array}{c} #1 \end{array}\!\! \right]} } % coordinate vector

%%%% - matrix

\newcommand{\Mat}[2]{ {\mathrm{Mat}_{#1}\left(#2\right)} }

\newcommand{\MATa}[1]{ {\left[\! \begin{array}{c} #1 \end{array} \!\right]} }
\newcommand{\MATb}[1]{ {\left[\! \begin{array}{cc} #1 \end{array} \!\right]} }
\newcommand{\MATc}[1]{ {\left[\! \begin{array}{ccc} #1 \end{array} \!\right]} }
\newcommand{\MATd}[1]{ {\left[\! \begin{array}{cccc} #1 \end{array} \!\right]} }
\newcommand{\MATe}[1]{ {\left[\! \begin{array}{ccccc} #1 \end{array} \!\right]} }
\newcommand{\MATf}[1]{ {\left[\! \begin{array}{cccccc} #1 \end{array} \!\right]} }
\newcommand{\MATg}[1]{ {\left[\! \begin{array}{ccccccc} #1 \end{array} \!\right]} }
\newcommand{\BASIS}[1]{ {\left[\! \begin{array}{c} #1 \end{array} \!\right]} } % BASIS matirx

\newcommand{\Tr}{ \mathrm{Tr}\, } % Trace
\newcommand{\tr}{ \mathrm{tr} }   % normaliezed trace

%%%% - permutation

\newcommand{\PERMa}[1]{ {\left( \begin{array}{c} #1 \end{array} \right)} }
\newcommand{\PERMb}[1]{ {\left( \begin{array}{cc} #1 \end{array} \right)} }
\newcommand{\PERMc}[1]{ {\left( \begin{array}{ccc} #1 \end{array} \right)} }
\newcommand{\PERMd}[1]{ {\left( \begin{array}{cccc} #1 \end{array} \right)} }
\newcommand{\PERMe}[1]{ {\left( \begin{array}{ccccc} #1 \end{array} \right)} }
\newcommand{\PERMf}[1]{ {\left( \begin{array}{cccccc} #1 \end{array} \right)} }
\newcommand{\PERMg}[1]{ {\left( \begin{array}{ccccccc} #1 \end{array} \right)} }


% Morphism

\newcommand{\Mor}[2]{ {\mathrm{Mor}_{#1}(#2)} }

\newcommand{\Aut}[2]{ {\mathrm{Aut}_{#1}(#2)} }
\newcommand{\End}[2]{ {\mathrm{End}_{#1}(#2)} }
\newcommand{\Iso}[2]{ {\mathrm{Iso}_{#1}(#2)} }
\newcommand{\Hom}[2]{ {\mathrm{Hom}_{#1}(#2)} }


\newcommand{\Inn}[2]{ {\mathrm{Inn}_{#1}\left(#2\right)} } % inner automorphism group
\newcommand{\Out}[2]{ {\mathrm{Out}_{#1}\left(#2\right)} } % outer automorphism group


%%% image - coimage - kernel - cokernel

\newcommand{\ch}{ \mathrm{ch} }         % characteristic

%\newcommand{\Im}{ {\mathrm{Im}\,} }
\newcommand{\im}{ {\mathrm{im}\,} }
\newcommand{\Image}{ {\mathrm{Im}\,} }

\newcommand{\Coim}{ {\mathrm{Coim}\,} }
\newcommand{\coim}{ {\mathrm{coim}\,} }
\newcommand{\Coimage}{ {\mathrm{Coim}\,} }
\newcommand{\coimage}{ {\mathrm{coim}\,} }

\newcommand{\Ker}{ {\mathrm{Ker}\,} }
%\newcommand{\ker}{ {\mathrm{ker}\,} }

\newcommand{\Coker}{ {\mathrm{Coker}\,} }
%\newcommand{\coker}{ {\mathrm{coker}\,} }
\newcommand{\Cok}{ {\mathrm{Cok}\,} }
%\newcommand{\cok}{ {\mathrm{cok}\,} }

%

\newcommand{\DD}{ \mathbb{D} }
\newcommand{\FF}{ \mathbb{F} }



\newcommand{\acts}{ : }  % left action
\newcommand{\acted}{ : } % right action

\newcommand{\Lact}[1]{ {\displaystyle \mathop{\curvearrowright}^{#1} } }
\newcommand{\Ract}[1]{ {\displaystyle \mathop{\curvearrowleft}^{#1} }  }

\newcommand{\actR}[1]{ {\displaystyle \mathop{\curvearrowright}^{#1} } }
\newcommand{\actL}[1]{ {\displaystyle \mathop{\curvearrowleft}^{#1} }  }

\newcommand{\tensor}[1]{ {\displaystyle \mathop{\otimes}_{#1} } }

\newcommand{\SG}[1]{ {\mathfrak{S}_{#1}} } % symmetric group
\newcommand{\AG}[1]{ {\mathfrak{A}_{#1}} } % alternating group
\newcommand{\CG}[1]{ {\mathfrak{C}_{#1}} } % cyclic group with order #1
\newcommand{\DG}[1]{ {\mathfrak{D}_{#1}} } % dihedral group with order 2 #1
\newcommand{\BSG}[1]{ {\widehat{\mathfrak{S}}_{#1}} } % binary symmetric group
\newcommand{\BAG}[1]{ {\widehat{\mathfrak{A}}_{#1}} } % binary alternating group
\newcommand{\BCG}[1]{ {\widehat{\mathfrak{S}}_{#1}} } % binary cyclic group with order 2 #1
\newcommand{\BDG}[1]{ {\widehat{\mathfrak{D}}_{#1}} } % binary dihedral group with order 4 #1



%Wely group

\newcommand{\sgn}{\mathrm{sgn}\,}



%Lie group

\newcommand{\AGL}[2]{ {\mathrm{AGL}_{#1} \left( #2 \right)} } % subset of Mat(#1 +1)
\newcommand{\ASL}[2]{ {\mathrm{ASL}_{#1} \left( #2 \right)} } % subset of Mat(#1 +1)


\newcommand{\Isom}[2]{ {\mathrm{Isom}_{#1}(#2)} }




\newcommand{\GL}[2]{ {\mathrm{GL}_{#1}(#2)} }
\newcommand{\SL}[2]{ {\mathrm{SL}_{#1}(#2)} }
\newcommand{\GU}[2]{ {\mathrm{U}_{#1}(#2)} }
\newcommand{\SU}[2]{ {\mathrm{SU}_{#1}(#2)} }
\newcommand{\GO}[2]{ {\mathrm{O}_{#1}(#2)} }
\newcommand{\SO}[2]{ {\mathrm{SO}_{#1}(#2)} }
\newcommand{\Gpin}[2]{ {\mathrm{pin}_{#1}(#2)} }
\newcommand{\pin}[2]{{ \mathrm{pin}_{#1} (#2) }}
\newcommand{\Spin}[2]{ {\mathrm{Spin}_{#1}(#2)} }
\newcommand{\GS}[2]{ {\mathrm{Sp}_{#1}(#2)} }
%\newcommand{\SS}[2]{ {\mathrm{SU}_{#1}(#2)} }

\newcommand{\PGL}[2]{ {\mathrm{PGL}_{#1}(#2)} }
\newcommand{\PSL}[2]{ {\mathrm{PSL}_{#1}(#2)} }
\newcommand{\PGU}[2]{ {\mathrm{PU}_{#1}(#2)} }
\newcommand{\PSU}[2]{ {\mathrm{PSU}_{#1}(#2)} }
\newcommand{\PGO}[2]{ {\mathrm{PO}_{#1}(#2)} }
\newcommand{\PSO}[2]{ {\mathrm{PSO}_{#1}(#2)} }
\newcommand{\PS}[2]{ {\mathrm{PSp}_{#1}(#2)} }
\newcommand{\PSp}[2]{ {\mathrm{PSp}_{#1}(#2)} }
\newcommand{\PGSp}[2]{ {\mathrm{PSp}_{#1}(#2)} }


%Lie algebra

\newcommand{\gl}[2]{ {\mathrm{gl}_{#1}(#2)} }
\newcommand{\Lsl}[2]{ {\mathrm{sl}_{#1}(#2)} }

\newcommand{\isom}[2]{ {\mathrm{isom}_{#1}(#2)} }

\newcommand{\Lsu}[2]{ {\mathrm{su}_{#1}(#2)} }
\newcommand{\gu}[2]{ {\mathrm{u}_{#1}(#2)} }
\newcommand{\su}[2]{ {\mathrm{su}_{#1}(#2)} }
%\newcommand{\go}[2]{ {\mathrm{o}_{#1}(#2)} }
\newcommand{\so}[2]{ {\mathrm{so}_{#1}(#2)} }
\newcommand{\gs}[2]{ {\mathrm{sp}_{#1}(#2)} }
%\newcommand{\ss}[2]{ {\mathrm{su}_{#1}(#2)} }
