Symmetrization and antisymmetrization of tensor products












4














How do I achieve something like



enter image description here



or



enter image description here



in LaTeX? The curved lines indicate symmetrization with respect to the indices they join, and the straight lines indicate antisymmetrization.






math-mode tensor







share|improve this question



























    4














    How do I achieve something like



    enter image description here



    or



    enter image description here



    in LaTeX? The curved lines indicate symmetrization with respect to the indices they join, and the straight lines indicate antisymmetrization.






    math-mode tensor







    share|improve this question

























      4












      4








      4


      1





      How do I achieve something like



      enter image description here



      or



      enter image description here



      in LaTeX? The curved lines indicate symmetrization with respect to the indices they join, and the straight lines indicate antisymmetrization.






      math-mode tensor







      share|improve this question













      How do I achieve something like



      enter image description here



      or



      enter image description here



      in LaTeX? The curved lines indicate symmetrization with respect to the indices they join, and the straight lines indicate antisymmetrization.






      math-mode tensor




      math-mode tensor






      share|improve this question













      share|improve this question











      share|improve this question




      share|improve this question










      asked Dec 9 at 23:41









      leastaction

      203111




      203111






















          2 Answers
          2






          active

          oldest

          votes


















          4














          Here is a simpleminded proposal.



          documentclass{article}
          
usepackage{tikz}
          
usetikzlibrary{tikzmark}
          
begin{document}
          
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
          
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
          
begin{tikzpicture}[overlay,remember picture]
          
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
          
draw[blue,thick] (j1) -- ++(0,-1.5ex)  -| (k1);
          
end{tikzpicture}
          
end{document}


          enter image description here



          As for your comments:



          documentclass{article}
          
usepackage{tikz}
          
usetikzlibrary{tikzmark}
          
begin{document}
          
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
          
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
          
begin{tikzpicture}[overlay,remember picture]
          
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
          
draw[blue,thick] (j1) -- ([yshift=-2pt]v1.south -| j1)  -| (k1);
          
end{tikzpicture}
          
quad
          
$u^{tikzmarknode{i2}{i}tikzmarknode{j2}{j}}
          
tikzmarknode{v2}{v}^{tikzmarknode{k2}{k}tikzmarknode{l2}{ell}}$
          
begin{tikzpicture}[overlay,remember picture]
          
draw[blue,thick] (j2.north east) to[out=30,in=150] (k2.north west);
          
draw[blue,thick] (i2) -- ([yshift=-2pt]v2.south -| i2)  -| (l2);
          
end{tikzpicture}
          
quad
          
$u^{tikzmarknode{i3}{i}tikzmarknode{j3}{j}}
          
tikzmarknode{v3}{v}^{tikzmarknode{k3}{k}tikzmarknode{l3}{ell}}$
          
begin{tikzpicture}[overlay,remember picture]
          
draw[blue,thick,rounded corners=2pt] (j3) --
          
([yshift=-2pt]v3.south -| j3) -| (k3);
          
draw[blue,thick] (i3.north) -- ++(0,3pt)  -| (l3);
          
end{tikzpicture}
          
end{document}


          enter image description here






          share|improve this answer



















          • 1




            Thanks! In your code, I suppose v1 isn’t being used.
            – leastaction
            Dec 10 at 0:00










          • Can you perhaps edit your answer to include the symmetrization below and antisymmetrization above as well? I couldn't get them to work in spite of using something like "draw[red,thick] (j1.south west) to [out=300,in=225] (k1.south west);"
            – leastaction
            Dec 10 at 2:35






          • 1




            @leastaction Sorry, was hibernating. Added some more options.
            – marmot
            Dec 10 at 6:07










          • Thanks. I meant to ask if it is possible to swap the curve with the |__| (i.e. the curve below, and the |---| above).
            – leastaction
            Dec 10 at 6:08






          • 1




            @leastaction I added a proposal. If you want things to be like in the last example, one may actually use Wick contractions for that, i.e. one may modify the package commands to give you that as well.
            – marmot
            Dec 10 at 6:20



















          2














          Also with pstricks (compile with xelatex):



          documentclass{article}
          
usepackage{pst-node}
          

          
begin{document}
          

          
$u^{rnode{i}{i}rnode{j}{j}} v^{rnode{k}{k}rnode{l}{ell}}$ 
          
ncarc[linecolor=blue,nodesep=1pt,arcangle=60]{i}{l}
          
ncbar[linecolor=blue,nodesep=1pt,arm=3pt,angle=-90]{j}{k}
          

          
end{document}



          enter image description here







          share|improve this answer























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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            4














            Here is a simpleminded proposal.



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ++(0,-1.5ex)  -| (k1);
            
end{tikzpicture}
            
end{document}


            enter image description here



            As for your comments:



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ([yshift=-2pt]v1.south -| j1)  -| (k1);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i2}{i}tikzmarknode{j2}{j}}
            
tikzmarknode{v2}{v}^{tikzmarknode{k2}{k}tikzmarknode{l2}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (j2.north east) to[out=30,in=150] (k2.north west);
            
draw[blue,thick] (i2) -- ([yshift=-2pt]v2.south -| i2)  -| (l2);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i3}{i}tikzmarknode{j3}{j}}
            
tikzmarknode{v3}{v}^{tikzmarknode{k3}{k}tikzmarknode{l3}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick,rounded corners=2pt] (j3) --
            
([yshift=-2pt]v3.south -| j3) -| (k3);
            
draw[blue,thick] (i3.north) -- ++(0,3pt)  -| (l3);
            
end{tikzpicture}
            
end{document}


            enter image description here






            share|improve this answer



















            • 1




              Thanks! In your code, I suppose v1 isn’t being used.
              – leastaction
              Dec 10 at 0:00










            • Can you perhaps edit your answer to include the symmetrization below and antisymmetrization above as well? I couldn't get them to work in spite of using something like "draw[red,thick] (j1.south west) to [out=300,in=225] (k1.south west);"
              – leastaction
              Dec 10 at 2:35






            • 1




              @leastaction Sorry, was hibernating. Added some more options.
              – marmot
              Dec 10 at 6:07










            • Thanks. I meant to ask if it is possible to swap the curve with the |__| (i.e. the curve below, and the |---| above).
              – leastaction
              Dec 10 at 6:08






            • 1




              @leastaction I added a proposal. If you want things to be like in the last example, one may actually use Wick contractions for that, i.e. one may modify the package commands to give you that as well.
              – marmot
              Dec 10 at 6:20
















            4














            Here is a simpleminded proposal.



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ++(0,-1.5ex)  -| (k1);
            
end{tikzpicture}
            
end{document}


            enter image description here



            As for your comments:



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ([yshift=-2pt]v1.south -| j1)  -| (k1);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i2}{i}tikzmarknode{j2}{j}}
            
tikzmarknode{v2}{v}^{tikzmarknode{k2}{k}tikzmarknode{l2}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (j2.north east) to[out=30,in=150] (k2.north west);
            
draw[blue,thick] (i2) -- ([yshift=-2pt]v2.south -| i2)  -| (l2);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i3}{i}tikzmarknode{j3}{j}}
            
tikzmarknode{v3}{v}^{tikzmarknode{k3}{k}tikzmarknode{l3}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick,rounded corners=2pt] (j3) --
            
([yshift=-2pt]v3.south -| j3) -| (k3);
            
draw[blue,thick] (i3.north) -- ++(0,3pt)  -| (l3);
            
end{tikzpicture}
            
end{document}


            enter image description here






            share|improve this answer



















            • 1




              Thanks! In your code, I suppose v1 isn’t being used.
              – leastaction
              Dec 10 at 0:00










            • Can you perhaps edit your answer to include the symmetrization below and antisymmetrization above as well? I couldn't get them to work in spite of using something like "draw[red,thick] (j1.south west) to [out=300,in=225] (k1.south west);"
              – leastaction
              Dec 10 at 2:35






            • 1




              @leastaction Sorry, was hibernating. Added some more options.
              – marmot
              Dec 10 at 6:07










            • Thanks. I meant to ask if it is possible to swap the curve with the |__| (i.e. the curve below, and the |---| above).
              – leastaction
              Dec 10 at 6:08






            • 1




              @leastaction I added a proposal. If you want things to be like in the last example, one may actually use Wick contractions for that, i.e. one may modify the package commands to give you that as well.
              – marmot
              Dec 10 at 6:20














            4












            4








            4






            Here is a simpleminded proposal.



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ++(0,-1.5ex)  -| (k1);
            
end{tikzpicture}
            
end{document}


            enter image description here



            As for your comments:



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ([yshift=-2pt]v1.south -| j1)  -| (k1);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i2}{i}tikzmarknode{j2}{j}}
            
tikzmarknode{v2}{v}^{tikzmarknode{k2}{k}tikzmarknode{l2}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (j2.north east) to[out=30,in=150] (k2.north west);
            
draw[blue,thick] (i2) -- ([yshift=-2pt]v2.south -| i2)  -| (l2);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i3}{i}tikzmarknode{j3}{j}}
            
tikzmarknode{v3}{v}^{tikzmarknode{k3}{k}tikzmarknode{l3}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick,rounded corners=2pt] (j3) --
            
([yshift=-2pt]v3.south -| j3) -| (k3);
            
draw[blue,thick] (i3.north) -- ++(0,3pt)  -| (l3);
            
end{tikzpicture}
            
end{document}


            enter image description here






            share|improve this answer














            Here is a simpleminded proposal.



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ++(0,-1.5ex)  -| (k1);
            
end{tikzpicture}
            
end{document}


            enter image description here



            As for your comments:



            documentclass{article}
            
usepackage{tikz}
            
usetikzlibrary{tikzmark}
            
begin{document}
            
$u^{tikzmarknode{i1}{i}tikzmarknode{j1}{j}}
            
tikzmarknode{v1}{v}^{tikzmarknode{k1}{k}tikzmarknode{l1}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (i1.north east) to[out=30,in=150] (l1.north west);
            
draw[blue,thick] (j1) -- ([yshift=-2pt]v1.south -| j1)  -| (k1);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i2}{i}tikzmarknode{j2}{j}}
            
tikzmarknode{v2}{v}^{tikzmarknode{k2}{k}tikzmarknode{l2}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick] (j2.north east) to[out=30,in=150] (k2.north west);
            
draw[blue,thick] (i2) -- ([yshift=-2pt]v2.south -| i2)  -| (l2);
            
end{tikzpicture}
            
quad
            
$u^{tikzmarknode{i3}{i}tikzmarknode{j3}{j}}
            
tikzmarknode{v3}{v}^{tikzmarknode{k3}{k}tikzmarknode{l3}{ell}}$
            
begin{tikzpicture}[overlay,remember picture]
            
draw[blue,thick,rounded corners=2pt] (j3) --
            
([yshift=-2pt]v3.south -| j3) -| (k3);
            
draw[blue,thick] (i3.north) -- ++(0,3pt)  -| (l3);
            
end{tikzpicture}
            
end{document}


            enter image description here







            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited Dec 10 at 6:17

























            answered Dec 9 at 23:54









            marmot

            86.3k499184




            86.3k499184








            • 1




              Thanks! In your code, I suppose v1 isn’t being used.
              – leastaction
              Dec 10 at 0:00










            • Can you perhaps edit your answer to include the symmetrization below and antisymmetrization above as well? I couldn't get them to work in spite of using something like "draw[red,thick] (j1.south west) to [out=300,in=225] (k1.south west);"
              – leastaction
              Dec 10 at 2:35






            • 1




              @leastaction Sorry, was hibernating. Added some more options.
              – marmot
              Dec 10 at 6:07










            • Thanks. I meant to ask if it is possible to swap the curve with the |__| (i.e. the curve below, and the |---| above).
              – leastaction
              Dec 10 at 6:08






            • 1




              @leastaction I added a proposal. If you want things to be like in the last example, one may actually use Wick contractions for that, i.e. one may modify the package commands to give you that as well.
              – marmot
              Dec 10 at 6:20














            • 1




              Thanks! In your code, I suppose v1 isn’t being used.
              – leastaction
              Dec 10 at 0:00










            • Can you perhaps edit your answer to include the symmetrization below and antisymmetrization above as well? I couldn't get them to work in spite of using something like "draw[red,thick] (j1.south west) to [out=300,in=225] (k1.south west);"
              – leastaction
              Dec 10 at 2:35






            • 1




              @leastaction Sorry, was hibernating. Added some more options.
              – marmot
              Dec 10 at 6:07










            • Thanks. I meant to ask if it is possible to swap the curve with the |__| (i.e. the curve below, and the |---| above).
              – leastaction
              Dec 10 at 6:08






            • 1




              @leastaction I added a proposal. If you want things to be like in the last example, one may actually use Wick contractions for that, i.e. one may modify the package commands to give you that as well.
              – marmot
              Dec 10 at 6:20








            1




            1




            Thanks! In your code, I suppose v1 isn’t being used.
            – leastaction
            Dec 10 at 0:00




            Thanks! In your code, I suppose v1 isn’t being used.
            – leastaction
            Dec 10 at 0:00












            Can you perhaps edit your answer to include the symmetrization below and antisymmetrization above as well? I couldn't get them to work in spite of using something like "draw[red,thick] (j1.south west) to [out=300,in=225] (k1.south west);"
            – leastaction
            Dec 10 at 2:35




            Can you perhaps edit your answer to include the symmetrization below and antisymmetrization above as well? I couldn't get them to work in spite of using something like "draw[red,thick] (j1.south west) to [out=300,in=225] (k1.south west);"
            – leastaction
            Dec 10 at 2:35




            1




            1




            @leastaction Sorry, was hibernating. Added some more options.
            – marmot
            Dec 10 at 6:07




            @leastaction Sorry, was hibernating. Added some more options.
            – marmot
            Dec 10 at 6:07












            Thanks. I meant to ask if it is possible to swap the curve with the |__| (i.e. the curve below, and the |---| above).
            – leastaction
            Dec 10 at 6:08




            Thanks. I meant to ask if it is possible to swap the curve with the |__| (i.e. the curve below, and the |---| above).
            – leastaction
            Dec 10 at 6:08




            1




            1




            @leastaction I added a proposal. If you want things to be like in the last example, one may actually use Wick contractions for that, i.e. one may modify the package commands to give you that as well.
            – marmot
            Dec 10 at 6:20




            @leastaction I added a proposal. If you want things to be like in the last example, one may actually use Wick contractions for that, i.e. one may modify the package commands to give you that as well.
            – marmot
            Dec 10 at 6:20











            2














            Also with pstricks (compile with xelatex):



            documentclass{article}
            
usepackage{pst-node}
            

            
begin{document}
            

            
$u^{rnode{i}{i}rnode{j}{j}} v^{rnode{k}{k}rnode{l}{ell}}$ 
            
ncarc[linecolor=blue,nodesep=1pt,arcangle=60]{i}{l}
            
ncbar[linecolor=blue,nodesep=1pt,arm=3pt,angle=-90]{j}{k}
            

            
end{document}



            enter image description here







            share|improve this answer




























              2














              Also with pstricks (compile with xelatex):



              documentclass{article}
              
usepackage{pst-node}
              

              
begin{document}
              

              
$u^{rnode{i}{i}rnode{j}{j}} v^{rnode{k}{k}rnode{l}{ell}}$ 
              
ncarc[linecolor=blue,nodesep=1pt,arcangle=60]{i}{l}
              
ncbar[linecolor=blue,nodesep=1pt,arm=3pt,angle=-90]{j}{k}
              

              
end{document}



              enter image description here







              share|improve this answer


























                2












                2








                2






                Also with pstricks (compile with xelatex):



                documentclass{article}
                
usepackage{pst-node}
                

                
begin{document}
                

                
$u^{rnode{i}{i}rnode{j}{j}} v^{rnode{k}{k}rnode{l}{ell}}$ 
                
ncarc[linecolor=blue,nodesep=1pt,arcangle=60]{i}{l}
                
ncbar[linecolor=blue,nodesep=1pt,arm=3pt,angle=-90]{j}{k}
                

                
end{document}



                enter image description here







                share|improve this answer














                Also with pstricks (compile with xelatex):



                documentclass{article}
                
usepackage{pst-node}
                

                
begin{document}
                

                
$u^{rnode{i}{i}rnode{j}{j}} v^{rnode{k}{k}rnode{l}{ell}}$ 
                
ncarc[linecolor=blue,nodesep=1pt,arcangle=60]{i}{l}
                
ncbar[linecolor=blue,nodesep=1pt,arm=3pt,angle=-90]{j}{k}
                

                
end{document}



                enter image description here








                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited Dec 10 at 1:29

























                answered Dec 10 at 1:16









                AboAmmar

                32.9k22882




                32.9k22882






























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