GrassmannCalculus`
ComposeCospan  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the book 4-dimensional vector space.
In[2]:=
★A;
;
 | ★ℬ |
4
The 3-cospan of the above four dimensional space is just the basis vectors but with signs.
In[3]:=
| ComposeCospan |
e
1
e
2
e
3
e
4
Out[3]=
{,-,,-}
e
4
e
3
e
2
e
1
This can also be entered in the following manner.
In[4]:=
3
| ★ |
e
1
e
2
e
3
e
4
Out[4]=
{,-,,-}
e
4
e
3
e
2
e
1
The 1-cospan is
In[5]:=
1
| ★ |
e
1
e
2
e
3
e
4
Out[5]=
{⋀⋀,-(⋀⋀),⋀⋀,-(⋀⋀)}
e
2
e
3
e
4
e
1
e
3
e
4
e
1
e
2
e
4
e
1
e
2
e
3
The exterior product of the k-spans and k-cospans give back copies of the original space. Remember that the various Grassmann product are Listable.
In[6]:=
| ★ |
1
e
1
e
2
e
3
e
4
1
| ★ |
e
1
e
2
e
3
e
4
 | ★ |
Out[6]=
{,,,}
e
1
e
2
e
3
e
4
Out[6]=
{⋀⋀,-(⋀⋀),⋀⋀,-(⋀⋀)}
e
2
e
3
e
4
e
1
e
3
e
4
e
1
e
2
e
4
e
1
e
2
e
3
Out[6]=
{⋀⋀⋀,⋀⋀⋀,⋀⋀⋀,⋀⋀⋀}
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
In[7]:=
| ★ |
2
e
1
e
2
e
3
e
4
2
| ★ |
e
1
e
2
e
3
e
4
| ★ |
Out[7]=
{⋀,⋀,⋀,⋀,⋀,⋀}
e
1
e
2
e
1
e
3
e
1
e
4
e
2
e
3
e
2
e
4
e
3
e
4
Out[7]=
{⋀,-(⋀),⋀,⋀,-(⋀),⋀}
e
3
e
4
e
2
e
4
e
2
e
3
e
1
e
4
e
1
e
3
e
1
e
2
Out[7]=
{⋀⋀⋀,⋀⋀⋀,⋀⋀⋀,⋀⋀⋀,⋀⋀⋀,⋀⋀⋀}
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
The following are the 1-span and 1-cospan of a subspace.
In[8]:=
| ★ |
1
e
1
e
3
1
| ★ |
e
1
e
3
| ★ |
Out[8]=
{,}
e
1
e
3
Out[8]=
{,-}
e
3
e
1
Out[8]=
{⋀,⋀}
e
1
e
3
e
1
e
3
The following forms will give all the k-cospans of a space.
In[9]:=
| ComposeCospan |
e
1
e
3
| ★ |
| ★ |
e
1
e
3
Out[9]=
{{⋀},{,-},{1}}
e
1
e
3
e
3
e
1
Out[9]=
{{⋀},{,-},{1}}
e
1
e
3
e
3
e
1
The following generates a list of specified k-cospans, here the even ones.
In[10]:=
 | ★ |
 | ★ |
e
1
e
2
e
3
e
4
Out[10]=
{ e 1 e 2 e 3 e 4 |
{ e 3 e 4 e 2 e 4 e 2 e 3 e 1 e 4 e 1 e 3 e 1 e 2 |
{1} |
The following generates the 2-cospan of a product composed of symbolic vectors.
In[11]:=
2
| ★ |
Out[11]=
{z,-y,x}
The following forms generate symbolic vectors for a 3-dimensional subspace and adds them to the VectorSymbols list.
In[12]:=
2
| ★ |
Out[12]=
{,-,}
s
3
s
2
s
1
In[13]:=
| ★ |
| ★ |
Out[13]=
{{⋀,-(⋀),⋀},{1}}
s
2
s
3
s
1
s
3
s
1
s
2
 |
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