Grassmann Algebra

ExpTheGeneralizedProduct.nb 5
Α m ����� Λ �Β k
� 0 Min�m, k� �Λ�Max�m, k�
This result, in which a sum of terms is identically zero, is a source of many useful identities
between exterior and interior products. It will be explored in Section 10.4 below.
Case Λ = Max[m, k]: Reduction to zero
10.5
When Λ is equal to the larger of the grades of the factors, the generalized product reduces to a
single interior product which is zero by virtue of its left factor being of lesser grade than its right
factor.
Α m ����� Λ �Β k
Case Λ > Max[m, k]: Undefined
� 0 Λ�Max�m, k�
When the order of the product Λ is greater than the larger of the grades of the factors of the
product, the generalized product cannot be expanded in terms of either of its factors and is
therefore undefined.
Α m ����� Λ �Β k
� Undefined Λ�Max�m, k�
10.6
10.7
10.3 The Symmetric Form of the Generalized Product
Expansion of the generalized product in terms of both factors
If Α and Β are both simple we can express the generalized Grassmann product more
m k
symmetrically by converting the interior products with the Interior Common Factor Theorem
[6.28].
To show this, we start with the definition [10.1] of the generalized Grassmann product.
Α m ����� Λ �Β k
� k
Λ �
� �
j�1
�Α ���� Β
m j
��Β
Λ
j
k�Λ
From the Interior Common Factor Theorem we can write the interior product Α m ���� Β j
2001 4 26
Λ
as: