Symmetrization and antisymmetrization of tensor products

How do I achieve something like

enter image description here

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

asked Dec 9 at 23:41

leastaction

203111

add a comment |

How do I achieve something like

enter image description here

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

asked Dec 9 at 23:41

leastaction

203111

add a comment |

How do I achieve something like

enter image description here

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

asked Dec 9 at 23:41

leastaction

203111

How do I achieve something like

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

asked Dec 9 at 23:41

leastaction

203111

asked Dec 9 at 23:41

leastaction

203111

asked Dec 9 at 23:41

leastaction

203111

asked Dec 9 at 23:41

leastaction

203111

asked Dec 9 at 23:41

leastaction

203111

add a comment |

2 Answers
2

active

oldest

votes

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

edited Dec 10 at 6:17

answered Dec 9 at 23:54

marmot

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

add a comment |

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}

edited Dec 10 at 1:29

answered Dec 10 at 1:16

AboAmmar

32.9k22882

add a comment |

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

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

edited Dec 10 at 6:17

answered Dec 9 at 23:54

marmot

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

add a comment |

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

edited Dec 10 at 6:17

answered Dec 9 at 23:54

marmot

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

add a comment |

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

edited Dec 10 at 6:17

answered Dec 9 at 23:54

marmot

86.3k499184

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

edited Dec 10 at 6:17

answered Dec 9 at 23:54

marmot

86.3k499184

edited Dec 10 at 6:17

answered Dec 9 at 23:54

marmot

86.3k499184

answered Dec 9 at 23:54

marmot

86.3k499184

answered Dec 9 at 23:54

marmot

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

add a comment |

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

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

@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

@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

add a comment |

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}

edited Dec 10 at 1:29

answered Dec 10 at 1:16

AboAmmar

32.9k22882

add a comment |

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}

edited Dec 10 at 1:29

answered Dec 10 at 1:16

AboAmmar

32.9k22882

add a comment |

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}

edited Dec 10 at 1:29

answered Dec 10 at 1:16

AboAmmar

32.9k22882

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}

edited Dec 10 at 1:29

answered Dec 10 at 1:16

AboAmmar

32.9k22882

edited Dec 10 at 1:29

answered Dec 10 at 1:16

AboAmmar

32.9k22882

answered Dec 10 at 1:16

AboAmmar

32.9k22882

answered Dec 10 at 1:16

AboAmmar

32.9k22882

add a comment |

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