Grassmann Algebra

TheInteriorProduct.nb 12
������ ��� �
�
����Γ
������
�  ��� �
�p
k �
����Γ
�
����� ������
Α � Βk ��� � Γ
�p
m � p
���������
we get the Interior Common Factor Formula.
����Γ
� Α
�p
m
Finally, by replacing
����� �����
Α � Βk with Αm � Β
m
k
In this case Γ p
����Γ
�p
�
���� Α��� �
m�
����Γ
�
���� ��� �
�p
k�
����Γ
�p
���� �Α m � Β k
� �����Γ
� p
is simple, the expression is a p-element but p = m+k.
The Interior Common Factor Theorem
The Common Factor Theorem enabled an explicit expression for a regressive product to be
derived. We now derive the interior product version called the Interior Common Factor
Theorem.
We start with the Common Factor Theorem in the form:
Α m � Β s
Ν
�
� � ���Αi
�
� Β��� � Αi
�m�p
s�
p
i�1
where Α �Α1 � Α1 �Α2 � Α2 � � �ΑΝ�
ΑΝ , Α is simple, p = m+sÐn, and Ν�� m m�p p m�p p
m�p p m m
p �.
�����
Suppose now that Β � Β , k = nÐs = mÐp. Substituting for Βs and using formula 6.23 allows us
s k
to write the term in brackets as
Αi � Β
m�p s
� Αi
k
�����
� Β � �Αi ���� Β� 1
k k k
�����
so that the Common Factor Theorem becomes:
Α m �Α1
k
Α m ���� Β k
Ν
� �
i�1
�Αi
k
���� Β��Αi k m�k
� Α1 �Α2�
Α2 � � �ΑΝ�
ΑΝ
m�k k m�k
k m�k
where k £ m, and Ν�� m
k �, and Α is simple.
m
This formula is a source of many useful relations in Grassmann algebra. It indicates that an
interior product of a simple element with another, not necessarily simple, element of equal or
lower grade, may be expressed in terms of the factors of the simple element of higher grade.
When Α m is not simple, it may always be expressed as the sum of simple components:
Α �Α1 �Αm 2 �Αm 3 � � (the i in the Αm
i are superscripts, not powers). From the linearity of
m m
2001 4 5
6.27
6.28