Grassmann Algebra

ExpTheGeneralizedProduct.nb 19
ToInteriorProductsB�1����� �� 2 3
Β1 � Β2 � Β3 �Β2 � Β1 � Β3 �Β3 � Β1 � Β2
Example: 4-elements
Alternatively we can use the GrassmannAlgebra function InteriorSum.
InteriorSum�1��� 4
Β1 � Β2 � Β3 � Β4 �Β2 � Β1 � Β3 � Β4 �Β3 � Β1 � Β2 � Β4 �Β4 � Β1 � Β2 � Β3
InteriorSum�2��� 4
2 �Β1 � Β2 � Β3 � Β4� � 2 �Β1 � Β3 � Β2 � Β4� � 2 �Β1 � Β4 � Β2 � Β3�
InteriorSum�3��� 4
Β1 � Β2 � Β3 � Β4 �Β1 � Β2 � Β4 � Β3 �Β1 � Β3 � Β4 � Β2 �Β2 � Β3 � Β4 � Β1
We can verify that these expressions are indeed zero by converting them to their scalar product
form. For example:
ToScalarProducts�InteriorSum�3���� 4
0
10.7 The Zero Generalized Sum
The zero generalized sum conjecture
The zero generalized sum conjecture conjectures that if the interior product operation is
replaced in the Zero Interior Sum Theorem by a generalized product of order other than zero
(that is, other than by the exterior product), then the theorem still holds. That is
2001 4 26
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p
10.16