Grassmann Algebra

ExpTheGeneralizedProduct.nb 34
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10.14 The Generalized Product of Intersecting
Orthogonal Elements
The case Λ < p
Consider three simple elements Α, Β and Γ . The elements Γ � Α and Γ � Β may then be
m k p
p m p k
considered elements with an intersection Γ. As has been shown in Section 10.12, generalized
p
products of such intersecting elements are zero whenever the grade Λ of the product is less than
the grade of the intersection. Hence the product will be zero for Λ < p, independent of the
orthogonality relationships of its factors.
� The case Λ ≥ p
Consider now the case of three simple elements Α, Β and Γ where Γ is totally orthogonal to
m k p p
both Α and Β (and hence to Α � Β). A simple element Γ is totally orthogonal to an element Α if
m k
m k
p
m
and only if Α ���� Γi � 0 for all Γi belonging to Γ. m p
The generalized product of the elements Γ � Α and Γ � Β
p m p k
of order Λ ≥ p is a scalar factor Γ ���� Γ
p p
times the generalized product of the factors Α and Β, but of an order lower by the grade of the
m k
common factor; that is, of order Λ–p.
Γ p
����Γ
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� Α�������� �
�p
m�
Λ �
�
�Γ1� Γ2 � � � Γp
Note here that the factors of Γ p
factors of Α. Nor the factors of Β. m k
���à p
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k�
�
�
���à p
�
���� Γ�����Α� ���� �Β� p�
m Λ�p k
���� Γi � 0 Λ�p
Α m ���� Γi �Β k
are not necessarily orthogonal to each other. Neither are the
We can test this out by making a table of cases. Here we look at the first 25 cases.
2001 4 26
10.39