I am reading the book by Loring W. Tu, `An introduction to manifolds'.
on page 23, he defines the permutation action on multilinear functions as
$$ (\sigma f)(v_{1}, v_{2}, \ldots, v_{k} ) = f(v_{\sigma(1)}, v_{\sigma(2)}, \ldots, v_{\sigma(k)}) . $$
He verified that this is indeed an action of the permutation group $S_k$ on multilinear function. That is, he veried that $(\tau\sigma) f = \tau(\sigma f )$.
I can follow the proof step by step, but I am still a little bit uneasy with the way of action.
Why is it $\sigma$, but not '$\sigma^{-1}$', on the right hand side?
We know that if a group $G$ acts on a set $M$, then the induced action of $G$ on the functions on $M$ is
$$ (g F)(x) = F(g^{-1}x) .$$
Here on the RHS, we have $g^{-1}$.