Grassmann Algebra

ExpTheGeneralizedProduct.nb 25
� The triple generalized sum
Triple generalized products
We define a triple generalized product to be a product either of the form ���x�����
�y
��������
�z or of the
�m
Λ k�
Μ p
form x����� �
m Λ ���y�����
�z
���.
Since we have shown above that these forms will in general be different, we
�k
Μ p�
will refer to the first one as the A form, and to the second as the B form.
Signed triple generalized products
We define the signed triple generalized product of form A to be the triple generalized product of
form A multiplied by the factor ��1� sA , where sA �
m Λ� 1 ���� Λ �1 �Λ�� �k � m� Μ� 2 1 ���� Μ �1 �Μ�.
2
We define the signed triple generalized product of form B to be the triple generalized product of
form B multiplied by the factor ��1�sB , where
sB � m Λ� 1 ���� 2 Λ �1 �Λ� � k Μ� 1 ���� 2 Μ �1 �Μ�.
Triple generalized sums
We define a triple generalized sum of form A to be the sum of all the signed triple generalized
products of form A of the same elements and which have the same grade n.
n
�
Λ�0
���1� sA � �
�
��x m ����� Λ �y k
����
����
�
n�Λ �z p
where sA � m Λ� 1 ���� Λ �Λ�1� � �m � k� �n �Λ�� 2 1 ���� �n �Λ� �n �Λ�1�.
2
We define a triple generalized sum of form B to be the sum of all the signed triple generalized
products of form B of the same elements and which have the same grade n.
n
�
Λ�0
���1� sB �xm ����� �
Λ ���y�
����
�k
n�Λ �z p
���
�
where sB � m Λ� 1 ���� 2 Λ �Λ�1� � k �n �Λ�� 1 ���� 2 �n �Λ� �n �Λ�1�.
For example, we tabulate below the triple generalized sums of form A for grades of 0, 1 and 2:
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Table��i, TripleGeneralizedSumA�i��x,y,z��, �i, 0, 2�� ��
m k p
TableForm
0
���x
� y
���
� z
�m
0 k�
0 p
1�m
1 ��1�
���
�
���x
� y
���
� z
���
1�k�m
� ��1�
���
�m
1 k�
0 p�
�
���x
� y
���
� z
���
�m
0 k�
1 p�
2 � ���
�
���x
� y
���
� z
���
k
� ��1�
���
�m
2 k�
0 p�
�
���x
� y
���
� z
���
�
�m
1 k�
1 p�
���x
� y
���
� z
�m
0 k�
2 p