Exterior Power
The kth exterior power of an element alpha in an exterior algebra LambdaV is given by the wedge product of alpha with itself k times. Note that if alpha has odd degree, then any higher power of alpha must be zero. The situation for even degree forms is different. For example, if
| alpha=e_1 ^ e_2+e_3 ^ e_4+e_5 ^ e_6, |
(1)
|
then
alpha^2 = 2e_1 ^ e_2 ^ e_3 ^ e_4+2e_1 ^ e_2 ^ e_5 ^ e_6+2e_3 ^ e_4 ^ e_5 ^ e_6
(2)
alpha^3 = 6e_1 ^ e_2 ^ e_3 ^ e_4 ^ e_5 ^ e_6
(3)
alpha^4 = 0.
(4)
See also
Exterior Algebra, Wedge ProductThis entry contributed by Todd Rowland
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Rowland, Todd. "Exterior Power." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExteriorPower.html