GrassmannCalculus`
ExteriorToRegressive  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
The following shows the effect of ExteriorToRegressive on the GrassmannBases in two different dimensions.
In[2]:=
★A;
[%]
 | GrassmannBases |
 | ExteriorToRegressive |
Out[2]=
{{1},{,,},{⋀,⋀,⋀},{⋀⋀}}
e
1
e
2
e
3
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
Out[2]=
{1},{,,},⋁,⋁,⋁,⋁⋁
e
1
e
2
e
3
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
In[3]:=
 | ★ℬ |
4
| GrassmannBases |
| ExteriorToRegressive |
Out[3]=
{{1},{,,,},{⋀,⋀,⋀,⋀,⋀,⋀},{⋀⋀,⋀⋀,⋀⋀,⋀⋀},{⋀⋀⋀}}
e
1
e
2
e
3
e
4
e
1
e
2
e
1
e
3
e
1
e
4
e
2
e
3
e
2
e
4
e
3
e
4
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
3
e
4
e
2
e
3
e
4
e
1
e
2
e
3
e
4
Out[3]=
{1},{,,,},⋁,⋁,⋁,⋁,⋁,⋁,-⋁⋁,-⋁⋁,-⋁⋁,-⋁⋁,⋁
e
1
e
2
e
3
e
4
e
1
e
2
e
1
e
3
e
1
e
4
e
2
e
3
e
2
e
4
e
3
e
4
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
3
e
4
e
2
e
3
e
4
e
1
-⋁⋁
e
2
e
3
e
4
Here is an expression involving the exterior product of a p- and q-element in a 3-space. Converting its exterior product to a regressive product necessarily introduces some complement operations.
In[4]:=
★A;X=1+⋀
[X]
j
p
k
q
| ExteriorToRegressive |
Out[4]=
1+⋀
j
p
k
q
Out[4]=
1+⋁
j
p
k
q
You can also convert expressions involving multigraded symbols, but they will be expanded into sums of terms involving graded symbols where necessary in order properly to compute the signs of the individual terms (whose signs depend on their grades). Here we perform a conversion in a 2-space.
In[5]:=
| ★ℬ |
2
u
{0,1,2,3}
v
{0,1,2,3}
j
{0,1,2,3}
k
{0,1,2,3}
 | ExteriorToRegressive |
Out[5]=
u
{0,1,2,3}
v
{0,1,2,3}
j
{0,1,2,3}
k
{0,1,2,3}
Out[5]=
u
{0,1,2,3}
v
{0,1,2,3}
j
0
k
0
j
0
k
1
j
0
k
2
j
1
k
0
j
1
k
1
j
2
k
0
Of course this can be simplified, but the simplification removes the conversion in the ⋀ term.
j
1
k
1
In[6]:=
Xs=
[Xr]
| ★ |
Out[6]=
u
{0,1,2,3}
v
{0,1,2,3}
j
0
k
0
j
1
k
0
j
2
k
0
j
0
k
1
j
0
k
2
j
1
k
1
You can apply again to get a final simplified result in this case.
ExteriorToRegressive
In[7]:=
| ExteriorToRegressive |
Out[7]=
u
{0,1,2,3}
v
{0,1,2,3}
j
1
k
1
j
0
k
0
j
1
k
0
j
2
k
0
j
0
k
1
j
0
k
2
In[8]:=
Clear[X,Xr,Xs]
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