In Kobayashi & Nomizu, the interior derivative of an r-form is defined as $\iota_X \omega = C(X \otimes \omega)$, where $C$ is the contraction associated with the pair $(1,1)$ and $\omega$ is interpreted as a tensor of type $(0,r)$. They then claim that \begin{equation*} (\iota_X)(Y_1, \dots, Y_{r-1}) = r \; \omega(X, Y_1, \dots, Y_{r-1}) \end{equation*} My question is: where does the factor of $r$ come from? This seems to be inconsistent with their previous conventions for forms.
Since they define $\omega_1 \wedge \omega_2 = \frac{1}{2}(\omega_1 \otimes \omega_2 - \omega_2 \otimes \omega_1)$, we should have \begin{equation*} C(X \otimes (\omega_1 \wedge \omega_2))(Y) = \frac{1}{2}( \omega_1(X) \omega_2(Y) - \omega_2(X) \omega_1(Y)) \end{equation*} whereas, using the determinant convention in K&N, given by \begin{equation*} \omega_1 \wedge \dots \wedge \omega_n = \frac{1}{n!} \det(\omega_i(X_j)) \end{equation*} we have \begin{equation*} 2 (\omega_1 \wedge \omega_2)(X,Y) = 2 \frac{1}{2} (\omega_1(X) \omega_2(Y) - \omega_2(X) \omega_1(Y)) \end{equation*} and so the two expressions are not equal. Where does the problem arise?
