Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 18
B
��1 � x, y�, �u, 1 � v��
Now we apply GrassmannIntegerPower to obtain the fourth power of B.
GrassmannIntegerPower�B, 4� ��MatrixForm
�
1 � 4x� 6u� y � 4u� v � y � 8u� x � y 4 y� 6v� y � 6x� y � 4v� x � y
4u� 6u� v � 6u� x � 4u� v � x 1� 4v� 6u� y � 8u� v � y � 4u� x
However, on the way to calculating the fourth power, the function
GrassmannIntegerPower has remembered the values of the all the integer powers up to
the fourth, in this case the second, third and fourth. We can get immediate access to these by
adding the Dimension of the space as a third argument. Since the power of a matrix may
take on a different form in spaces of different dimensions (due to some terms being zero
because their degree exceeds the dimension of the space), the power is recalled only by
including the Dimension as the third argument.
Dimension
4
GrassmannIntegerPower�B, 2, 4� ��MatrixForm �� Timing
�0. Second, �
1 � 2x� u � y 2 y� v � y � x � y
2u� u � v � u � x 1� 2v� u � y ��
GrassmannIntegerPower�B, 3, 4� ��MatrixForm �� Timing
1 � 3x� 3u� y � u � v � y � 2u� x � y 3 y�3v� y � 3x� y
�0. Second, �
3u� 3u� v � 3u� x � u � v � x 1�3v� 3u� y � 2u� v
GrassmannIntegerPower�B, 4, 4� ��MatrixForm �� Timing
�0. Second, �
GrassmannMatrixPower
1 � 4x� 6u� y � 4u� v � y � 8u� x � y 4y� 6v� y � 6x�
4u� 6u� v � 6u� x � 4u� v � x 1� 4v� 6u� y � 8u�
The principal and more general function for calculating matrix powers provided by
GrassmannAlgebra is GrassmannMatrixPower. Here we verify that it gives the same
result as our simple GrassmannIntegerPower function.
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