Symmetrization and antisymmetrization of tensor products
up vote
4
down vote
favorite
How do I achieve something like
or
in LaTeX? The curved lines indicate symmetrization with respect to the indices they join, and the straight lines indicate antisymmetrization.
math-mode tensor
add a comment |
up vote
4
down vote
favorite
How do I achieve something like
or
in LaTeX? The curved lines indicate symmetrization with respect to the indices they join, and the straight lines indicate antisymmetrization.
math-mode tensor
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
How do I achieve something like
or
in LaTeX? The curved lines indicate symmetrization with respect to the indices they join, and the straight lines indicate antisymmetrization.
math-mode tensor
How do I achieve something like
or
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
asked Dec 9 at 23:41
leastaction
203111
203111
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
4
down vote
accepted
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}
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}
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 |
up vote
2
down vote
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}
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
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}
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}
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 |
up vote
4
down vote
accepted
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}
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}
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 |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
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}
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}
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}
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}
edited Dec 10 at 6:17
answered Dec 9 at 23:54
marmot
83.3k493178
83.3k493178
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
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
add a comment |
up vote
2
down vote
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}
add a comment |
up vote
2
down vote
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}
add a comment |
up vote
2
down vote
up vote
2
down vote
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}
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
32.1k22781
32.1k22781
add a comment |
add a comment |
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