Grassmann Algebra

ExpTheGeneralizedProduct.nb 3
For brevity in the rest of this book, where no confusion will arise, we may call the generalized
Grassmann product simply the generalized product.
10.2 Defining the Generalized Product
Definition of the generalized product
The generalized Grassmann product of order Λ of an m-element Α and a simple k-element Β
m k
denoted Α����� �Β and defined by
m Λ k
Α m ����� Λ �Β k
Β k
� Β 1
Λ
� k
Λ �
� �
j�1
� Β 1
k�Λ
�Α ���� Β
m j��Β
Λ
j
k�Λ
� Β 2
Λ
� Β 2
� �
k�Λ
is
10.1
Here, Β is expressed in all of the �
k
k
� essentially different arrangements of its 1-element factors
Λ
into a Λ-element and a (k-Λ)-element. This pairwise decomposition of a simple exterior product
is the same as that used in the Common Factor Theorem [3.28] and [6.28].
The grade of a generalized Grassmann product Α m ����� Λ �Β k
is therefore m + k - 2Λ, and like the
grades of the exterior and interior products in terms of which it is defined, is independent of the
dimension of the underlying space.
Note the similarity to the form of the Interior Common Factor Theorem. However, in the
definition of the generalized product there is no requirement for the interior product on the right
hand side to be an inner product, since an exterior product has replaced the ordinary
multiplication operator.
For brevity in the discussion that follows, we will assume without explicitly drawing attention
to the fact, that an element undergoing a factorization is necessarily simple.
Case Λ = 0: Reduction to the exterior product
When the order Λ of a generalized product is zero, Βj reduces to a scalar, and the product
Λ
reduces to the exterior product.
2001 4 26
Α m ����� 0 �Β k
� Α m � Β k
Λ�0
10.2