GrassmannCalculus`
ExteriorToInterior  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
The following shows the effect of ExteriorToInterior on the GrassmannBases in two different dimensions.
In[2]:=
★A;
[%]
 | GrassmannBases |
 | ExteriorToInterior |
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 |
| ExteriorToInterior |
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 an interior product necessarily introduces some complement operations.
In[4]:=
★A;X=1+⋀
[X]
j
p
k
q
| ExteriorToInterior |
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 (which 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}
 | ExteriorToInterior |
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
Note that if is applied to this result it will convert terms of the form ⊖ to (where
GrassmannSimplify
b
α
m
b(⊖)
1
α
m
b
is a scalar).In[6]:=
| ★ |
Out[6]=
u
{0,1,2,3}
v
{0,1,2,3}
j
1
k
1
1
k
1
j
0
1
k
2
j
0
j
0
k
0
j
1
k
0
j
2
k
0
In[7]:=
Clear[X,Xr]
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""