Grassmann Algebra

TheExteriorProduct.nb 18
BasisTable�1�
� BASIS
�
0
1
�
l
�e1�
BasisTable�4�
� BASIS
�
0
1
�
l
�e1, e2, e3, e4�
�
2
�e1 � e2, e1� e3, e1� e4, e2� e3, e2� e4, e3� e4�
� 3 �e1 � e2 � e3, e1 � e2 � e4, e1 � e3 � e4, e2 � e3 � e4�
� 4
Standard ordering
e1 � e2 � e3 � e4
The standard ordering of the basis of � 1 is defined as the ordering of the elements in the
declared list of basis elements. This ordering induces a natural standard ordering on the basis
elements of � m : if the basis elements of � 1 were letters of the alphabet arranged alphabetically,
then the basis elements of � m would be words arranged alphabetically.
For example, if we take {A,B,C,D} as basis, we can see that the basis elements for each of the
bases obtained from entering BasisTable[] are arranged alphabetically. (Equivalently, if the
basis elements were digits, the ordering would be numeric.)
2001 4 5
DeclareBasis��A, B, C, D��
�A, B, C, D�
BasisTable��
� BASIS
�
0
1
�
l
�A, B, C, D�
�
2
�A � B, A � C, A � D, B � C, B � D, C � D�
� 3 �A � B � C, A � B � D, A � C � D, B � C � D�
� 4
A � B � C � D