Grassmann Algebra

TheExteriorProduct.nb 25
� Calculating determinants
As an example take the following matrix. We can calculate its determinant with Mathematica's
inbuilt Det function:
� a 0 0 0 0 0 0 0
��������������������������������
���������������������������������
0 0 0 c 0 0 0 a
0 b 0 0 0 f 0 0
0 0 0 0 e 0 0 0
Det�
�
0 0 e 0 0 0 g 0
0 f 0 d 0 c 0 0
g 0 0 0 0 0 0 h
� 0 0 0 0 d 0 b 0 �
ab 2 c 2 e 2 h � abce 2 f 2 h
Alternatively, we can set � 1 to be 8-dimensional.
DeclareBasis�8�
�e1, e2, e3, e4, e5, e6, e7, e8�
and then evaluate the exterior product of the 8 rows or columns expressed as 1-elements. Here
we use GrassmannSimplify (alias �) to expand by columns.
����a e1 � ge7���be3 � fe6���ee5���ce2 � de6��
�ee4 � de8���fe3 � ce6���ge5 � be8���ae2 � he7���
abce 2 �bc� f 2 � he1 � e2 � e3 � e4 � e5 � e6 � e7 � e8
The determinant is then the coefficient of the basis 8-element.
Speeding up calculations
Because of the large number of rules GrassmannSimplify has to check, it is often faster to
use the generalized Laplace expansion technique. That is, the order in which the products are
calculated is arranged with the objective of minimizing the number of resulting terms produced
at each stage. We do this by dividing the product up into groupings of factors where each
grouping contains as few basis elements as possible in common with the others. We then
calculate the result from each grouping before combining them into the final result. Of course,
we must ensure that the sign of the overall regrouping is the same as the original expression.
In the example below, we have rearranged the determinant calculation above into three
groupings, which we calculate successively before combining the results. This grouping took an
order of magnitude less time than the direct approach above.
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