张量积的对称化和反对称化

Question

我如何实现类似

或者

在 LaTeX 中？曲线表示相对于它们所连接的指标的对称性，直线表示反对称性。

Answer 1

这是一个简单的建议。

\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}

至于您的评论：

\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}

Answer 2

同样使用 pstricks (使用 xelatex 编译)：

\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}