Grassmann Algebra

ExpTheGeneralizedProduct.nb 10
� Example: 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. This leads however to a sum of terms which
is zero.
Α m ����� Λ �Β k
� 0 Min�m, k� �Λ�Max�m, k�
When Λ is equal to the larger of the grades of the factors, the generalized product reduces to a
single interior product which is zero by virtue of its left factor being of lesser grade than its right
factor. Suppose Λ = k > m. Then:
Α m ����� k �Β k
� Α���� Β � 0 Λ�k � m
m k
These relationships are the source of an interesting suite of identities relating exterior and
interior products. We take some examples; in each case we verify that the result is zero by
converting the expression to its scalar product form.
Example 1
A123 � ToInteriorProductsB�Α����� �Β� 2 3
Α � Β1 � Β2 � Β3 �Α� Β2 � Β1 � Β3 �Α� Β3 � Β1 � Β2
ToScalarProducts�A123�
0
Example 2
A235 � ToInteriorProductsB�Α����� �Β� 2 3 5
Α1 � Α2 � Β1 � Β2 � Β3 � Β4 � Β5 �Α1 � Α2 � Β1 � Β3 � Β2 � Β4 � Β5 �
Α1 � Α2 � Β1 � Β4 � Β2 � Β3 � Β5 �Α1 � Α2 � Β1 � Β5 � Β2 � Β3 � Β4 �
Α1 � Α2 � Β2 � Β3 � Β1 � Β4 � Β5 �Α1 � Α2 � Β2 � Β4 � Β1 � Β3 � Β5 �
Α1 � Α2 � Β2 � Β5 � Β1 � Β3 � Β4 �Α1 � Α2 � Β3 � Β4 � Β1 � Β2 � Β5 �
Α1 � Α2 � Β3 � Β5 � Β1 � Β2 � Β4 �Α1 � Α2 � Β4 � Β5 � Β1 � Β2 � Β3
Expand�ToScalarProducts�A235��
0
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