GrassmannCalculus`
RegressiveToExterior  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
The following shows the effect of RegressiveToExterior on various expressions composed from the GrassmannBases in two different dimensions.
In[2]:=
★A;{⋁,⋀⋁⋀,⋀⋀⋁⋀}
[%]
e
1
e
2
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
e
3
 | RegressiveToExterior |
Out[2]=
{⋁,⋀⋁⋀,⋀⋀⋁⋀}
e
1
e
2
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
e
3
Out[2]=
⋀,⋀⋀⋀,⋀⋀⋀⋀
e
1
e
2
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
e
3
In[3]:=
 | ★ℬ |
4
e
1
e
2
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
e
3
e
1
e
2
e
3
e
1
e
3
e
4
| RegressiveToExterior |
Out[3]=
{⋁,⋀⋁⋀,⋀⋀⋁⋀,⋀⋀⋁⋀⋀}
e
1
e
2
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
e
3
e
1
e
2
e
3
e
1
e
3
e
4
Out[3]=
⋀,⋀⋀⋀,-⋀⋀⋀⋀,⋀⋀⋀⋀⋀
e
1
e
2
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
e
3
e
1
e
2
e
3
e
1
e
3
e
4
Here is an expression involving the exterior product of a p- and q-element in a 3-space. Converting its regressive product to an exterior product necessarily introduces some complement operations.
In[4]:=
★A;X=1+⋁
[X]
j
p
k
q
| RegressiveToExterior |
Out[4]=
1+⋁
j
p
k
q
Out[4]=
1+⋀
j
p
k
q
Note that is equal to .
(m+k)(★n-(m+k))
(-1)
m(★n-m)+k(★n-k)
(-1)
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}
 | RegressiveToExterior |
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
1
k
2
j
2
k
0
j
2
k
1
j
2
k
2
Note that if is applied to this result it will convert the terms back to interior products (since they are deemed somewhat simpler).
GrassmannSimplify
In[6]:=
| ★ |
Out[6]=
j
0
k
2
j
1
k
1
j
1
k
2
j
2
k
1
j
2
k
2
u
{0,1,2,3}
v
{0,1,2,3}
j
2
1
k
0
In[7]:=
Clear[X,Xr]
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