Grassmann Algebra

ExpTheGeneralizedProduct.nb 20
� Generating the zero generalized sum
In GrassmannAlgebra we can use GeneralizedSum[p,Λ][Β1 � Β2 � Β3 � �] to generate
the zero generalized sum expression. Here p is the grade of the first factor. We obtain an
equivalent sum if p is replaced by p–Λ since for each i:
Β i
����� �Β
k�p Λ i
p
� ��1� �p����k�p�� � i
p
����� Λ �Β i
k�p
GeneralizedSum�2, 1��Β1 � Β2 � Β3 � Β4�
Β1 � Β2 � 1 Β3 � Β4 �Β1 � Β3 � 1 Β2 � Β4 �Β1 � Β4 � 1 Β2 � Β3 �
Β2 � Β3 � 1 Β1 � Β4 �Β2 � Β4 � 1 Β1 � Β3 �Β3 � Β4 � 1 Β1 � Β2
We can use the graded variable Β as an argument to get the same result.
4
GeneralizedSum�2, 1��� 4
Β1 � Β2 � 1 Β3 � Β4 �Β1 � Β3 � 1 Β2 � Β4 �Β1 � Β4 � 1 Β2 � Β3 �
Β2 � Β3 � 1 Β1 � Β4 �Β2 � Β4 � 1 Β1 � Β3 �Β3 � Β4 � 1 Β1 � Β2
� Exploring the conjecture
We can use GrassmannAlgebra and Mathematica to explore the conjecture by tabulating cases
to see if the result is zero. Here we tabulate the first 50 cases.
Flatten�Table�ToScalarProducts�GeneralizedSum�p, ��x k ��,
�k, 2, 8�, �p, 1, k � 1�, �Λ, 1,Min�p, k � p����
�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, 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�
10.8 Nilpotent Generalized Products
Nilpotent products of simple elements
In the case Β k
2001 4 26
is equal to Α m and Α m is simple and m is not zero, form B becomes:
�
Α����� �Α � �
m Λ m k
Λ �
j�1
�Α � Α
m j
� ���� Αj
m�Λ Λ