In Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces"

we have the formula for the exterior derivative

$(p+1)d\omega(X_{1}, \ldots X_{p+1})=\Sigma_{i=1}^{p+1} (-1)^{i+1} X_{i} \cdot \omega(X_{1}, \ldots , \hat{X_{i}}, \ldots X_{p+1})+\Sigma_{i<j} (-1)^{i+j}\omega([X_{i}, X_{j}], X_{1}, \ldots , \hat{X_{i}}, \ldots \hat{X_{j}}, \ldots , X_{p+1})$

but in other sources on-line they have the same formula without the factor of $p+1$ on the left-hand side.

Are there two different definitions of the exterior derivative, or is there an error in Helgason?


I am just making a guess. There are two approches to wedge product.

  1. From wiki $$(\omega\wedge\eta)(X_1,\dots ,X_{n+m}):=\frac{1}{n!m!}\sum_{\sigma\in S(n+m)}\epsilon(\sigma)\omega(X_{\sigma(1)},\dots,X_{\sigma(n)})\cdot\eta(X_{\sigma(n+1)},\dots,X_{\sigma(n+m)})$$
  2. From Helgason $$(\omega\wedge\eta)(X_1,\dots ,X_{n+m}):=\frac{1}{(n+m)!}\sum_{\sigma\in S(n+m)}\epsilon(\sigma)\omega(X_{\sigma(1)},\dots,X_{\sigma(n)})\cdot\eta(X_{\sigma(n+1)},\dots,X_{\sigma(n+m)}).$$ These differ only by the factor ($\frac{1}{(n+m)!}$ changes with $\frac{1}{n!m!}$). However in both cases $d$ can be defined axiomatic. In wiki and in Helgason by theorem 2.5. But in axiomatic approaches there are wedge products (which differ). So my GUESS is: that is the reason why $d$ differs in these sources.
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