Wedge products, graded commutativity, and associativity.

A reference for this material is Chapter 14 of John M. Lee. Introduction to smooth manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013. xvi+708pp. ISBN: 978-1-4419-9981-8.

After renormalization, the wedge product is the antisymmetrization of the tensor product

\[ \omega \wedge \eta = \frac {(k + \ell ) !}{k!\, \ell ! } A_{k + \ell } (\omega \otimes \eta ) . \]
Solution:

Let \(\tau \in S_{k + \ell }\) denote the permutation

\[ 1 \mapsto k + 1 , \ldots , \ell \mapsto k + \ell , \ell + 1 \mapsto 1 , \ldots , \ell + k \mapsto k . \]
Note that \(\text {sign}(\tau ) = (-1)^{k \ell }\). Then
\begin{align*} k! \, \ell ! & \, \omega \wedge \eta (X_1, \ldots , X_{k + \ell }) \\ & = ( k + \ell ) !\, A_{k + \ell } (\omega \otimes \eta ) (X_1, \ldots , X_{k + \ell })\\ & = \sum _{\sigma \in S_{k + \ell }} \text {sign}(\sigma )\, \omega ( X_{\sigma (1) } , \ldots , X_{\sigma (k)})\, \eta ( X_{\sigma (k+1)} , \ldots , X_{\sigma (k + \ell )})\\ & = \sum _{\sigma \in S_{k + \ell }} \text {sign}(\sigma )\, \eta ( X_{\sigma \tau (1 )} , \ldots , X_{\sigma \tau ( \ell )}) \, \omega ( X_{\sigma \tau (\ell + 1) } , \ldots , X_{\sigma \tau (\ell + k)}) \\ & = \text {sign}(\tau ) \sum _{\sigma \in S_{k + \ell }} \text {sign}(\sigma )\, \eta ( X_{\sigma (1 )} , \ldots , X_{\sigma ( \ell )}) \, \omega ( X_{\sigma (\ell + 1) } , \ldots , X_{\sigma (\ell + k)}) \\ & = (-1)^{k \ell } k! \, \ell ! \, \eta \wedge \omega (X_1, \ldots , X_{k + \ell }). \end{align*}
Solution:
\begin{align*} (\omega _1 \wedge \omega _2) \wedge \omega _3 & = \dfrac {(k_1 + k_2)!}{k_1!\,k_2!} A_{k_1 + k_2} (\omega _1 \otimes \omega _2) \wedge A_{k_3} ( \omega _3 ) \\ & = \dfrac {(k_1 + k_2 + k_3 )!}{k_1!\,k_2!\, k_3!} A_{k_1 + k_2 + k _3} ( A_{k_1 + k_2} (\omega _1 \otimes \omega _2) \otimes A_{k_3} ( \omega _3 ) )\\ & = \dfrac {(k_1 + k_2 + k_3 )!}{k_1!\,k_2!\, k_3!} A_{k_1 + k_2 + k _3} ( \omega _1 \otimes \omega _2 \otimes \omega _3 ). \end{align*}

The last expression is associative, so the first one is.