Csdrhrt

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.

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}

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}