Grassmann Algebra

ExpTheGeneralizedProduct.nb 7
It is easy to see, by using the distributivity of the generalized product that this relationship holds
also for non-simple Α and Β. m k
Expansion in terms of the other factor
We can now prove the alternative expression [10.3] for Α����� �Β in terms of the factors of a simple
m Λ k
Α by expanding the right hand side of the quasi-commutativity relationship [10.9].
m
Α m ����� Λ �Β k
� ��1� �m����k�� � k
����� �Α � ��1�
Λ m �m�Λ���k�Λ� �
��
i�1
m
Λ �
�Β
k
���� Α i
Λ ��Αi
m�Λ
Interchanging the order of the terms of the exterior product then gives the required alternative
expression:
Α m ����� Λ �Β k
� m
Λ �
� �
i�1
Α i
m�Λ ��Β k
Α m �� Α 1
Λ
� Α1
m�Λ
���� Α i
� 0 �Λ�Min�m, k�
Λ
�� Α2
Λ
� Α2 �� �
m�Λ
10.4 Calculating with Generalized Products
� Entering a generalized product
In GrassmannAlgebra you can enter a generalized product in the form:
GeneralizedProduct���X, Y�
where Λ is the order of the product and X and Y are the factors. On entering this expression into
Mathematica it will display it in the form:
X � Λ Y
Alternatively, you can click on the generalized product button � � � on the �� palette, and
�
then tab through the placeholders, entering the elements of the product as you go.
To enter multiple products, click repeatedly on the button. For example to enter a concatenated
product of four elements, click three times:
������ � ������� � ������� � ��
If you select this expression and enter it, you will see how the inherently binary product is
grouped.
2001 4 26