Grassmann Algebra

TheInteriorProduct.nb 14
InteriorCommonFactorTheorem�Α 3 ���� Β�
Α 3 � Β��Α1 � Α2 � Α3 � Β��
�Β � Α3� Α1 � Α2 � �Β � Α2� Α1 � Α3 � �Β � Α1� Α2 � Α3
InteriorCommonFactorTheorem�Α 4 ���� Β�
Α 4 � Β��Α1 � Α2 � Α3 � Α4 � Β����Β � Α4� Α1 � Α2 � Α3 �
�Β � Α3� Α1 � Α2 � Α4 � �Β � Α2� Α1 � Α3 � Α4 � �Β � Α1� Α2 � Α3 � Α4
Β is a 2-element
� ��Αi � Αj� ���� Β
2
i,j
�Α1 � Α2 � � � Αm� ���� Β �
2
����1� i�j�1 �Α1 � � � ���� i � � � ���� j � � � Αm
InteriorCommonFactorTheorem�Α ���� Β� 3 2
Α � Β �� Α1 � Α2 � Α3 � Β
3 2
2
�Β 2
��
� Α2 � Α3� Α1 � �Β � Α1 � Α3� Α2 � �Β
2
2
InteriorCommonFactorTheorem�Α ���� Β� 4 2
Α � Β �� Α1 � Α2 � Α3 � Α4 � Β
4 2
2
�Β 2
�Β 2
Β is a 3-element
��
� Α1 � Α2� Α3
� Α3 � Α4� Α1 � Α2 � �Β � Α2 � Α4� Α1 � Α3 � �Β
2
2
� Α1 � Α4� Α2 � Α3 � �Β � Α1 � Α3� Α2 � Α4 � �Β
2
2
InteriorCommonFactorTheorem�Α ���� Β� 4 3
Α � Β �� Α1 � Α2 � Α3 � Α4 � Β
4 3
3
�Β 3
�� ��Β � Α2 � Α3 � Α4� Α1 �
3
� Α1 � Α3 � Α4� Α2 � �Β � Α1 � Α2 � Α4� Α3 � �Β
3
3
� Α2 � Α3� Α1 � Α4 �
� Α1 � Α2� Α3 � Α4
� Α1 � Α2 � Α3� Α4
� The list form of the Interior Common Factor Theorem
We start with the Interior Common Factor Theorem.
2001 4 5
6.32