TITLE: Is there a preferable convention for defining the wedge product?
QUESTION [50 upvotes]: There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)!}{k!l!}Alt(\alpha\otimes\beta)$,
where $\alpha$ and $\beta$ are any forms of degree $k$ and $l$ respectively, and $Alt(\cdot)$ take the alternating part of the tensor.
But, is there a rationale to prefer one of them among the others?
If not, what do you prefer? and for what reason?

REPLY [2 votes]: I prefer the convention as in Kobayashi-Nomizu, as this is the natural one when using the modern approach to algebraic structures. In this set-up, the exterior algebra of a vector space $V$ is the quotient $\bigwedge V = TV/I$, where $TV$ is the tensor algebra of $V$, and I is the bilateral ideal generated by elements of the form $x\otimes x$.
As discussed in the Appendix of the reference "Euclidean Clifford Algebra", in this situation, one should define
$
\alpha \wedge \beta := Alt (\alpha\otimes\beta)
$