2015-11-18-143925.sagews

[0, 1, 2, 3, 4, 5]

t = StandardTableaux(4).random_element()
G = DiGraph([[t,t]], loops=True)
latex(G)
view(G)

\begin{tikzpicture} \definecolor{cv0}{rgb}{0.0,0.0,0.0} \definecolor{cfv0}{rgb}{1.0,1.0,1.0} \definecolor{clv0}{rgb}{0.0,0.0,0.0} \definecolor{cv0v0}{rgb}{0.0,0.0,0.0} % \Vertex[style={minimum size=1.0cm,draw=cv0,fill=cfv0,text=clv0,shape=circle},LabelOut=false,L=\hbox{${\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}} \raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{1-3} \lr{1}&\lr{2}&\lr{4}\\\cline{1-3} \lr{3}\\\cline{1-1} \end{array}$} }$},x=2.5cm,y=2.5cm]{v0} % \Loop[dist=3.0cm,dir=NO,style={color=cv0v0,},](v0) % \end{tikzpicture}

d3-based renderer not yet implemented

Let $G=S_5$. Then

%gap
G:=SymmetricGroup(5);
Order(G);

Sym( [ 1 .. 5 ] ) 120

its order is as above.

%sage
from sage.combinat.matrices.hadamard_matrix import hadamard_matrix
hadamard_matrix?

File: /projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/combinat/matrices/hadamard_matrix.py
Signature : hadamard_matrix()
Docstring :
Tries to construct a Hadamard matrix using a combination of Paley
and Sylvester constructions.

EXAMPLES:

   sage: hadamard_matrix(12).det()
   2985984
   sage: 12^6
   2985984
   sage: hadamard_matrix(1)
   [1]
   sage: hadamard_matrix(2)
   [ 1  1]
   [ 1 -1]
   sage: hadamard_matrix(8)
   [ 1  1  1  1  1  1  1  1]
   [ 1 -1  1 -1  1 -1  1 -1]
   [ 1  1 -1 -1  1  1 -1 -1]
   [ 1 -1 -1  1  1 -1 -1  1]
   [ 1  1  1  1 -1 -1 -1 -1]
   [ 1 -1  1 -1 -1  1 -1  1]
   [ 1  1 -1 -1 -1 -1  1  1]
   [ 1 -1 -1  1 -1  1  1 -1]
   sage: hadamard_matrix(8).det() == 8^4
   True

We note that the method hadamard_matrix() returns a normalised
Hadamard matrix (the entries in the first row and column are all
+1)

   sage: hadamard_matrix(12)
   [ 1  1  1  1  1  1| 1  1  1  1  1  1]
   [ 1  1  1 -1 -1  1|-1 -1  1 -1 -1  1]
   [ 1  1  1  1 -1 -1|-1  1 -1  1 -1 -1]
   [ 1 -1  1  1  1 -1|-1 -1  1 -1  1 -1]
   [ 1 -1 -1  1  1  1|-1 -1 -1  1 -1  1]
   [ 1  1 -1 -1  1  1|-1  1 -1 -1  1 -1]
   [-----------------+-----------------]
   [ 1 -1 -1 -1 -1 -1|-1  1  1  1  1  1]
   [ 1 -1  1 -1 -1  1| 1 -1 -1  1  1 -1]
   [ 1  1 -1  1 -1 -1| 1 -1 -1 -1  1  1]
   [ 1 -1  1 -1  1 -1| 1  1 -1 -1 -1  1]
   [ 1 -1 -1  1 -1  1| 1  1  1 -1 -1 -1]
   [ 1  1 -1 -1  1 -1| 1 -1  1  1 -1 -1]

H.det().factor()

2^12