Grassmann Algebra

TheRegressiveProduct.nb 28
Similarly for Α2 we have the same formula, except that w replaces v.
�Ξ � Ψ���w � x � y � z� � �Ξ � w � x � y � z��Ψ��Ψ � w � x � y � z��Ξ
Again applying this to each of the terms of Α 2 gives:
Α2 � �v � w � x � y � z��w � �v � w � x � y � z��x �
�v � w � x � y � z��y � �v � w � x � y � z��z � w � x � y � z
Hence the factorization is congruent to:
Α1 � Α2 � �v � z���w � x � y � z�
Verification by expansion of this product shows that this is indeed the factorization required.
� Factorizing elements expressed in terms of basis elements
We now take the special case where the element to be factorized is expressed in terms of basis
elements. This will enable us to develop formulae from which we can write down the
factorization of an element (almost) by inspection.
The development is most clearly apparent from a specific example, but one that is general
enough to cover the general concept. Consider a 3-element in a 5-space, where we suppose the
coefficients to be such as to ensure the simplicity of the element.
Α 3 � a1 e1 � e2 � e3 � a2 e1 � e2 � e4 � a3 e1 � e2 � e5
�a4 e1 � e3 � e4 � a5 e1 � e3 � e5 � a6 e1 � e4 � e5
�a7 e2 � e3 � e4 � a8 e2 � e3 � e5 � a9 e2 � e4 � e5
�a10 e3 � e4 � e5
Choose Β1 � e1 , Β2 � e2 , Β3 � e3 , Β � e4 � e5 , then we can write each of the factors
2
Βi � Β 2
as a cobasis element.
Β1 � Β 2
Β2 � Β 2
Β3 � Β 2
� e1 � e4 � e5 � e2 � e3
�������������
� e2 � e4 � e5 ��e1 � e3
�������������
� e3 � e4 � e5 � e1 � e2
�������������
Consider the cobasis element e2 � e3 ������������� , and a typical term of Α 3 � e2 � e3
������������� , which we write as
�a ei � ej � ek��e2 � e3 ������������� . The Common Factor Theorem tells us that this product is zero if
ei � ej � ek does not contain e2 � e3 . We can thus simplify the product Α � e2 � e3
3 ������������� by
dropping out the terms of Α 3 which do not contain e2 � e3 . Thus:
Α � e2 � e3
3 ������������� � �a1 e1 � e2 � e3 � a7 e2 � e3 � e4 � a8 e2 � e3 � e5��e2 � e3 �������������
Furthermore, the Common Factor Theorem applied to a typical term �e2 � e3 � ei��e2 � e3 �������������
occur yields a 1-element congruent to the
of the expansion in which both e2 � e3 and e2 � e3 �������������
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