Grassmann Algebra

ExpTheGeneralizedProduct.nb 18
Β k
� Β 1
Λ
� k
Λ �
�
j�1
Β j
Λ
� Β 1
k�Λ
����  j
k�Λ
� Β 2
Λ
� 0
� Β2 � �
k�Λ
10.13
Note that this is the same result as in [10.11] and [10.12] except that the elements of grade Λ
have exchanged places with those of grade k–Λ in the interior products. We might express this
more mnemonically as:
� Β 1
Λ
Β
k
� Β 1
Λ
� Β 1
k�Λ
����  1
k�Λ
� Β 2
Λ
� Β2 Λ
The Zero Interior Sum Theorem
� Β 2
� Β
k�Λ
3
� Β
Λ
3
� �
k�Λ
����  2
k�Λ
� Β 2
Λ
���� Β3 � � � 0
k�Λ
10.14
These two results are collected together and referred to as the Zero Interior Sum Theorem. It is
valid for 0 < Λ < k.
� Β 1
Λ
Β
k
� Β1 Λ
� Β 1
k�Λ
� Β 1
k�Λ
����  1
k�Λ
����  1
Λ
� Β2 Λ
� Β2 Λ
� Β2 � Β
k�Λ
3 � Β
Λ
3 � �
k�Λ
����  2
k�Λ
� Β 2
���� Β
k�Λ
2
Λ
� Generating the zero interior sum
� Β 2
Λ
� Β 2
���� Β
k�Λ
3
Λ
���� Β3 � � � 0
k�Λ
� � � 0
We can generate the zero interior sum by applying ToInteriorProductsB to the
generalized product of an element with unity. For example:
Example: 2-elements
ToInteriorProductsB�1����� �� 1 2
0
Example: 3-elements
2001 4 26
ToInteriorProductsB�1����� �� 1 3
Β1 � Β2 � Β3 �Β1 � Β3 � Β2 �Β2 � Β3 � Β1
10.15