Grassmann Algebra

ExpTheGeneralizedProduct.nb 21
We can see from the right hand side that Α m ����� Λ �Α m is zero for all Λ except Λ equal to m (where the
generalized product reduces to the interior (and in this case inner) product). Thus
Α m ����� Λ �Α m � 0, Λ�m
10.17
For example, we can calculate the generalized products of all orders for the 3-element Α, to see
3
that they are all zero except for Λ equal to 3.
Table�ToInnerProducts�Α 3 ����� Λ �Α 3 �, �Λ, 0,3��
�0, 0, 0, Α1 � Α2 � Α3 � Α1 � Α2 � Α3�
This result gives rise to a series of relationships among the exterior and interior products of
factors of a simple element. These relationships are independent of the dimension of the space
concerned. For example, applying the result to simple 2, 3 and 4-elements gives the following
relationships at the interior product level:
Example: 2-elements
ToInteriorProducts�Α 2 ����� 1 �Α 2 � � 0
�Α1 � Α2 � Α1��Α2 � �Α1 � Α2 � Α2��Α1 �� 0
Example: 3-elements
ToInteriorProducts�Α 3 ����� 1 �Α 3 � � 0
�Α1 � Α2 � Α3 � Α1��Α2 � Α3 �
�Α1 � Α2 � Α3 � Α2��Α1 � Α3 � �Α1 � Α2 � Α3 � Α3��Α1 � Α2 �� 0
ToInteriorProducts�Α 3 ����� 2 �Α 3 � � 0
�Α1 � Α2 � Α3 � Α1 � Α2��Α3 �
�Α1 � Α2 � Α3 � Α1 � Α3��Α2 � �Α1 � Α2 � Α3 � Α2 � Α3��Α1 �� 0
Example: 4-elements
2001 4 26
ToInteriorProducts�Α 4 ����� 1 �Α 4 � � 0
�Α1 � Α2 � Α3 � Α4 � Α1��Α2 � Α3 � Α4 �
�Α1 � Α2 � Α3 � Α4 � Α2��Α1 � Α3 � Α4 �
�Α1 � Α2 � Α3 � Α4 � Α3��Α1 � Α2 � Α4 �
�Α1 � Α2 � Α3 � Α4 � Α4��Α1 � Α2 � Α3 �� 0