Grassmann Algebra

ExpTheGeneralizedProduct.nb 30
10.12 The Generalized Product of Intersecting
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. Generalized products of such intersecting elements
p
are zero whenever the grade Λ of the product is less than the grade of the intersection.
����Γ
�p
�
� Α�������� �
m�
Λ �
�
���à p
�
� ��� � 0 �p
k�
We can test this by reducing the expression to scalar products and tabulating cases.
Flatten�Table�A � ToScalarProducts� ����Γ
�
� Α�������� �
�p
m�
Λ �
�
Print���m, k, p, �, A��; A,�m, 0, 3�,
�k, 0, 3�, �p, 1, 3�, �Λ, 0,p� 1���
���à p
�
� ����; k�
�0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0�
Of course, this result is also valid in the case that either or both the grades of Α m and Β k
Γ p
����� �
Λ ����Γ
�p
� The case Λ ≥ p
�
� ��� �
k�
�
�
���à p
�
� Α
m�
�������� Λ �Γ p
������ �à � 0 �p
p Λ p
is zero.
10.29
10.30
For the case where Λ is equal to, or greater than the grade of the intersection p, the generalized
product of such intersecting elements may be expressed as the interior product of the common
factor Γ with a generalized product of the remaining factors. This generalized product is of
p
order lower by the grade of the common factor. Hence the formula is only valid for Λ ≥ p (since
the generalized product has only been defined for non-negative orders).
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