Grassmann Algebra

TheRegressiveProduct.nb 41
ToProductFormula��Α � Β��x,n� m k 3
��1� m�n Α � Β � x1 � x2 � x3 �Α� x1 � Β � x2 � x3 �
m k
m k
Α � x2 � Β � x1 � x3 �Α� x3 � Β � x1 � x2 �
m k
m k
��1�m�n Α � x1 � x2 � Β � x3 � ��1�
m k
m�n Α � x1 � x3 � Β
m k
��1� m�n Α � x2 � x3 � Β � x1 �Α� x1 � x2 � x3 � Β
m k m k
� x2 �
This output is not particularly easy to read, as Mathematica adopts the precedence that the
exterior product has over the regressive product, and hence does not show the parentheses.
Replacing the parentheses and grouping terms manually gives:
�Α � Β���x1 � x2 � x3�
m k
� ��1� m�n Α m ��Β k
� x1 � x2 � x3�
��Α � x1���Β� x2 � x3� �
m k
�Α � x2���Β� x1 � x3� � �Α � x3���Β� x1 � x2�
m k
m k
���1� m�n ��� Α � x1 � x2���Β� x3� �
m k
� Α � x1 � x3���Β� x2� � � Α � x2 � x3���Β� x1��
m k
m k
The decomposition formula
��Α m � x1 � x2 � x3��
In the General Product Formula [3.43], putting k equal to nÐm permits the left-hand side to be
expressed as a scalar multiple of x and hence expresses a type of 'decomposition' of x into
p p
components.
Β k
p
x � � ���1�
p r�0
r��n�m� ����Α
�
� xi
��� ��  � xi�
Ν �m
p�r�
n�m r
���
�������������������������������� ��������������
i�1
�
x p � x1
r
� x1 � x2 � x2 � � � xΝ
p�r r p�r
r
�Α m � Β
n�m
� xΝ , Ν��
p�r
p
r �
The first and last terms of this sum, those for which r = 0 and r = p are:
2001 4 5
3.46
3.47