Grassmann Algebra

TheRegressiveProduct.nb 19
We can also express Α m and Β k
Α m � Α
m�p � Γ p
in terms of Γ p
Β �Γ
k p
� Β
k�p
by:
By starting with the regressive product Α � Β and performing a series of straightforward
m k
algebraic manipulations using these definitions, we obtain the result required. Summation is
always over the index i.
Α m � Β k
� ����
Α
�
m�p � Γ p
����
�
�
����Γ
�
�  ��� �
�p
k�p�
����
Α
�
� ����Α
�
�  ��� � à �
�m
k�p�
p
����Α
�
�  ��� �
�m
k�p�
�
�
��� ����ai�Α
�
� Β ��� � Αi
� m k�p�
p
m�p � Γ p
���� ai Αi
p
�
� Β ��� � Γ
k�p�
p
����
�
��� ����ai�Αi
� Αi � Β
� m�p p k�p
����
� Αi
� p
��� ����Αi
�
�m�p
�����
�
ai�Αi
���
�
� Β ��� � Αi ���
� p � k�p�
p
����Αi
�
� Γ � Β ��� � Αi
�m�p
p k�p�
p
��� ����Αi
�
� Β��� � Αi
�m�p
k�
p
In sum, we can write:
Ν
�
� � ���Αi
�
� Β��� � Αi
�m�p
k�
p
Α � Β
m k
i�1
Α �Α1 � Α1
m m�p p
�Α2 � Α2
m�p p
� � �ΑΝ�
ΑΝ
m�p p
p � m � k � n
An analogous formula may be obtained mutatis mutandis by factoring Β
k
must now be simple.
Β k
Α m � Β k
�Β1
p
Ν
�
� �
�
i�1
���Α � Βi
m k�p
����
� Βi
� p
� Β1 �Β2 � Β2 � � �ΒΝ�
ΒΝ
k�p p k�p
p k�p
�� m
p �
p � m � k � n
3.28
rather than Α m . Hence Β k
�� k
p �
It is evident that if both elements are simple, expansion in factors of the element of lower grade
will be the more efficient.
When neither Α m nor Β k
is simple, the Common Factor Rule can still be applied to the simple
component terms of the product.
Multiple regressive products may be treated by successive applications of the formulae above.
2001 4 5
3.29