Grassmann Algebra

TheRegressiveProduct.nb 31
To obtain a 1-element factor of the m-element
¥ Select an (mÐ1)-element belonging to at least one of the terms, for example y���z.
¥ Drop any terms not containing the selected (mÐ1)-element.
¥ Factor this (mÐ1)-element from the resulting expression and eliminate it.
¥ The 1-element remaining is a 1-element factor of the m-element.
To factorize the m-element
¥ Select an m-element belonging to at least one of the terms, for example x �y���z.
¥ Create m different (mÐ1)-elements by dropping a different 1-element factor each time. The
sign of the result is not important, since a scalar factor will be determined in the last step.
¥ Obtain m independent 1-element factors corresponding to each of these (mÐ1)-elements.
¥ The original m-element is congruent to the exterior product of these 1-element factors.
¥ Compare this product to the original m-element to obtain the correct scalar factor and hence
the final factorization.
Example 1: Factorizing a 2-element in a 4-space
Suppose we have a 2-element in a 4-space, and we wish to apply this algorithm to obtain a
factorization. We have already seen that such an element is in general not simple. We may
however use the preceding algorithm to obtain the simplicity conditions on the coefficients.
Α 2 � a1�e1 � e2 � a2�e1 � e3 � a3�e1 � e4 � a4�e2 � e3 � a5�e2 � e4 � a6�e3 � e4
¥ Select a 2-element belonging to at least one of the terms, say e1 � e2 .
¥ Drop e2 to create e1 . Then drop e1 to create e2 .
¥ Select e1 .
¥ Drop the terms a4�e2 � e3 � a5�e2 � e4 � a6�e3 � e4 since they do not contain e1 .
¥ Factor e1 from a1�e1 � e2 � a2�e1 � e3 � a3�e1 � e4 and eliminate it to give factor Α1 .
Α1 � a1�e2 � a2�e3 � a3�e4
¥ Select e2 .
¥ Drop the terms a2�e1 � e3 � a3�e1 � e4 � a6�e3 � e4 since they do not contain e2 .
¥ Factor e2 from a1�e1 � e2 � a4�e2 � e3 � a5�e2 � e4 and eliminate it to give factor Α2 .
Α2 ��a1�e1 � a4�e3 � a5�e4
¥ The exterior product of these 1-element factors is
Α1 � Α2 � �a1�e2 � a2�e3 � a3�e4���� a1�e1 � a4�e3 � a5�e4�
� a1 ���a1�e1
� e2 � a2 e1 � e3 � a3 e1 � e4 �
�
a4 e2 � e3 � a5 e2 � e4 � ���
�a3 a4 � a2 a5 ���
e3 � e4 �
��������������������������������� ��
� a1 � �
¥ Comparing this product to the original 2-element Α gives the final factorization as
2
2001 4 5
Α �
2 1
������� ��a1�e2 � a2�e3 � a3�e4���� a1�e1 � a4�e3 � a5�e4�
a1