使用矩阵的二项式树？

Question

我有针对特定组的以下代码，现在我希望以相同的方式显示每个排列的矩阵表示：

\documentclass[11pt]{article}
\usepackage{forest}
\begin{document}

\textit{Generator $(12)$}

\[
\begin{forest}
for tree={grow'=east,l sep=8em,s sep=3em,circle,inner sep=2pt,fill}
[,label=left:{$(12)$}
 [,label=above:{$()$},edge label={node[midway,sloped,above]{$(12)$}}
  [,label=right:$(12)$,edge label={node[midway,sloped,above]{$(12)$}}]
  [,label=right:$(13)$,edge label={node[midway,sloped,below]{$(13)$}}]
 ]
 [,label=below:{$(123)$},edge label={node[midway,sloped,below]{$(13)$}}
  [,label=right:$(23)$,edge label={node[midway,sloped,above]{$(12)$}}]
  [,label=right:$(12)$,edge label={node[midway,sloped,below]{$(13)$}}]
 ]
]
\end{forest}
\]
\end{document}

进一步的问题是，我该如何对齐这一点，以便两棵树的起点对齐。我想在它们之间画一个向右的箭头。

\textit{Generator $(132)$}
\[
\begin{forest}
for tree={grow'=east,l sep=5em,s sep=2em,circle,inner sep=2pt,fill}
[,label=left:{$(132)$}
 [,label=above:{$(12)$},edge label={node[midway,sloped,above]{$(23)$}}
  [,label=right:$(132)$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$(23)$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
 [,label=below:{$(123)$},edge label={node[midway,sloped,below]{$(132)$}}
  [,label=right:$(13)$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$()$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
]
\end{forest}
\rightarrow

\begin{forest}
for tree={grow'=east,l sep=7em,s sep=4em,circle,inner sep=2pt,fill}
[,label=left:{$\begin{bmatrix} 0 & 0 & 1 \\ 
1 & 0 & 0 \\ 
0 & 1 & 0 
\end{bmatrix}$}
 [,label=above:{$\begin{bmatrix} 0 & 1 & 0 \\ 
1 & 0 & 0 \\ 
0 & 0 & 1 
\end{bmatrix}$},edge label={node[midway,sloped,above]{$(23)$}}
  [,label=right:$\begin{bmatrix} 0 & 0 & 1 \\ 
1 & 0 & 0 \\ 
0 & 1 & 0 
\end{bmatrix}$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$\begin{bmatrix} 1 & 0 & 0 \\ 
0 & 0 & 1 \\ 
0 & 1 & 0 
\end{bmatrix}$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
 [,label=below:{$\begin{bmatrix} 0 & 1 & 0 \\ 
0 & 0 & 1 \\ 
1 & 0 & 0 
\end{bmatrix}$},edge label={node[midway,sloped,below]{$(132)$}}
  [,label=right:$\begin{bmatrix} 0 & 0 & 1 \\ 
0 & 1 & 0 \\ 
1 & 0 & 0 
\end{bmatrix}$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$\begin{bmatrix} 1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1 
\end{bmatrix}$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
]
\end{forest}
\]

Answer 1

我没有收到错误（当然是在加载之后amsmath）bmatrix。

\documentclass[11pt]{article}
\usepackage{forest}
\usepackage{amsmath}
\begin{document}

\textit{Generator $(12)$}

\[
\begin{forest}
for tree={grow'=east,l sep=8em,s sep=3em,circle,inner sep=2pt,fill}
[,label=left:{$(12)=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{bmatrix}$}
 [,label=above:{$()$},edge label={node[midway,sloped,above]{$(12)=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{bmatrix}$}}
  [,label=right:$(12)$,edge label={node[midway,sloped,above]{$(12)$}}]
  [,label=right:$(13)$,edge label={node[midway,sloped,below]{$(13)$}}]
 ]
 [,label=below:{$(123)$},edge label={node[midway,sloped,below]{$(13)$}}
  [,label=right:$(23)$,edge label={node[midway,sloped,above]{$(12)$}}]
  [,label=right:$(12)$,edge label={node[midway,sloped,below]{$(13)$}}]
 ]
]
\end{forest}
\]
\end{document}

（我可能看到一个离题错误：这些矩阵不是我所知道的任何置换矩阵，它们不是群元素，因为它们不可逆，但可能只是我不知道这种符号。）

至于附加要求：使用\vcenter{\hbox{...}}。（我想你可以问一个后续问题，这很快就完成了，我相信可以以\hspace*某种方式避免，但现在我不知道如何。）

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{forest}
\usepackage{pdflscape}
\begin{document}
\begin{landscape}
\textit{Generator $(12)$}
\[\hspace*{-10em}\vcenter{\hbox{
\begin{forest}
for tree={grow'=east,l sep=5em,s sep=2em,circle,inner sep=2pt,fill}
[,label=left:{$(132)$}
 [,label=above:{$(12)$},edge label={node[midway,sloped,above]{$(23)$}}
  [,label=right:$(132)$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$(23)$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
 [,label=below:{$(123)$},edge label={node[midway,sloped,below]{$(132)$}}
  [,label=right:$(13)$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$()$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
]
\end{forest}}}
\rightarrow\hspace*{-10em}
\vcenter{\hbox{\begin{forest}
for tree={grow'=east,l sep=7em,s sep=4em,circle,inner sep=2pt,fill}
[,label=left:{$\begin{bmatrix} 0 & 0 & 1 \\ 
1 & 0 & 0 \\ 
0 & 1 & 0 
\end{bmatrix}$}
 [,label=above:{$\begin{bmatrix} 0 & 1 & 0 \\ 
1 & 0 & 0 \\ 
0 & 0 & 1 
\end{bmatrix}$},edge label={node[midway,sloped,above]{$(23)$}}
  [,label=right:$\begin{bmatrix} 0 & 0 & 1 \\ 
1 & 0 & 0 \\ 
0 & 1 & 0 
\end{bmatrix}$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$\begin{bmatrix} 1 & 0 & 0 \\ 
0 & 0 & 1 \\ 
0 & 1 & 0 
\end{bmatrix}$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
 [,label=below:{$\begin{bmatrix} 0 & 1 & 0 \\ 
0 & 0 & 1 \\ 
1 & 0 & 0 
\end{bmatrix}$},edge label={node[midway,sloped,below]{$(132)$}}
  [,label=right:$\begin{bmatrix} 0 & 0 & 1 \\ 
0 & 1 & 0 \\ 
1 & 0 & 0 
\end{bmatrix}$,edge label={node[midway,sloped,above]{$(23)$}}]
  [,label=right:$\begin{bmatrix} 1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1 
\end{bmatrix}$,edge label={node[midway,sloped,below]{$(132)$}}]
 ]
]
\end{forest}}}
\]
\end{landscape}
\end{document}