Grassmann Algebra

ExpTheGeneralizedProduct.nb 4
Case 0 < Λ < Min[m, k]: Reduction to exterior and interior products
When the order Λ of the generalized product is greater than zero but less than the smaller of the
grades of the factors in the product, the product may be expanded in terms of either factor
(provided it is simple). The formula for expansion in terms of the first factor is similar to the
definition [10.1] which expands in terms of the second factor.
Α m ����� Λ �Β k
� m
Λ �
� �
i�1
Α i
m�Λ ��Β k
Α m �� Α 1
Λ
� Α1
m�Λ
���� Α i
� 0 �Λ�Min�m, k�
Λ
�� Α2
Λ
� Α2 �� �
m�Λ
In Section 10.3 we will prove this alternative form, showing in the process that the product can
be expanded in terms of both factors, thus underscoring the essential symmetry of the product.
Case Λ = Min[m, k]: Reduction to the interior product
When the order of the product Λ is equal to the smaller of the grades of the factors of the
product, there is only one term in the sum and the generalized product reduces to the interior
product.
Α m ����� Λ �Β k
Α m ����� Λ �Β k
� Α m ���� Β k
� Β���� Α
k m
Λ�k � m
Λ�m � k
10.3
10.4
This is a particularly enticing property of the generalized product. If the interior product of two
elements is non-zero, but they are of unequal grade, an interchange in the order of their factors
will give an interior product which is zero. If an interior product of two factors is to be non-zero,
it must have the larger factor first. The interior product is non-commutative. By contrast, the
generalized product form of the interior product of two elements of unequal grade is
commutative and equal to that interior product which is non-zero, whichever one it may be.
This becomes evident if we put Λ equal to k in the first of the formulae of the previous
subsection, and Λ equal to m in the second.
Case Min[m, k] < Λ < Max[m, k]: Reduction to zero
When the order of the product Λ is greater than the smaller of the grades of the factors of the
product, but less than the larger of the grades, the generalized product may still be expanded in
terms of the factors of the element of larger grade, leading however to a sum of terms which is
zero.
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