Grassmann Algebra

ExpTheGeneralizedProduct.nb 35
Flatten�
Table�ToScalarProducts� ����Γ
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m�
Λ ����Γ
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�p
k�
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���� Γ�����Α� ���� �Β���. �p
p�
m Λ�p k
OrthogonalSimplificationRules���Α, Γ�, �Β, Γ���, m p k p
�m, 0, 2�, �k, 0, m�, �p, 1, m � 1�, �Λ, p,p�Min�m, k����
�0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0�
By combining the results of the previous section with those of this section, we see that we can
also write that:
����Γ
�p
�
� Α
m�
�������� Λ �Β k
�Α����� �
m Λ ����Γ
�p
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���� Γ�����Α����� �Β� �
�p
p�
m Λ k
��1� p Λ � ����
�
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� Α�������� �
���� ���� à � ��1�
�p
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Λ k�
p
pm � � ���Α����� �
�m
Λ ����Γ
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� ��� �p
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���� Γ
� p
Γ �Γ1� Γ2 � � � Γp Α ���� Γi �Β���� Γi � 0
p
m k
We check the second rule by taking the first 25 cases.
10.40
Flatten�Table�
ToScalarProducts� ����Α�����
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� p
pm� ����Γ
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���� Γ�����Α����� �Β� � �.
�p
p�
m Λ k
OrthogonalSimplificationRules���Α, Γ�, �Β, Γ���, m p k p
�m, 0, 2�, �k, 0, m�, �p, 1, m � 1�, �Λ, 0,Min�m, k����
�0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0�
10.15 Generalized Products in Lower Dimensional
Spaces
Generalized products in 0, 1, and 2-spaces
In this section we summarize the properties of generalized products for spaces of dimension 0, 1
and 2.
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