Two slightly different definitions of wedge product of alternating tensors

Question

Let $\alpha$ and $\beta$ be alternating tensors of order $k$ and $l$ respectively. In some textbooks, their wedge product is defined as $$ (\alpha \wedge \beta)(v_1, \cdots, v_{k+l}) = \frac{1}{k! l!} \sum_\sigma \text{sgn}(\sigma) \, \alpha(v_{\sigma(1)}, \cdots, v_{\sigma(k)}) \, \beta (v_{\sigma(k+1)}, \cdots, v_{\sigma(k+l)}) $$ where $\sigma$ is a permutation of $1, \cdots, k+l$. But the factor $1 / k! l!$ does not appear in some other textbooks. In both case, the wedge product seems to be distributive, associative, and skew-commutative. Are there some important formulas that depend on the choice of the factor in the definition? Or don't I need to care about the choice at all?

Answer 1

Sometimes people use $\frac{(k+l)!}{k!l!}$ as well. There is no deep difference, what matters is that you are coherent with your notations.