Does a pseudoscalar commute or anti-commute with a vector?

Question

It seems to me the answer depends on the number of dimensions.

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Suppose 3D:

$$ I=e_1e_2e_3 $$

then

$$ e_1I = e_1e_1e_2e_3 = -e_1e_2e_1e_3 = e_1e_2e_3e_1=Ie_1 $$

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But if 2D:

$$ I=e_1e_2 $$

then

$$ e_1I=e_1e_1e_2=-e_1e_2e_1=-Ie_1 $$

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Is this correct, it depends on if the algebra is odd or even dimensions? I have read a couple of books on geometric algebra, but I never notice this distinction being mentioned. I am afraid I may be doing something incorrect here?

Answer 1

You are right that this depends on the dimension:

$$ v I = (-1)^{N-1} I v, $$

where $v$ is any vector, and the "unit" pseudoscalar $ I = e_1 \cdots e_N $ is constructed by some ordered product of orthonormal basis vectors.

I've definitely seen this in GA literature. I could be wrong, but think that "GA for physicists" was one such place.