Let $V$ be a finite dimensional vector space and $\mathcal T^k(V^*)$ be the set of all the covariant $k$-tensors on $V$.
The symmetric group $S_k$ acts on $\mathcal T^k(V^*)$ as follows: Given $\alpha\in \mathcal T^k(V^*)$ and $\sigma\in S_k$, we define $^\sigma\alpha\in \mathcal T^k(V^*)$ as:
$$^\sigma\alpha(v_1,\ldots, v_n)=\alpha(v_{\sigma(1)},\ldots, v_{\sigma(n)}), \quad \forall v_1,\ldots, v_n\in V$$
Here we are interpretting a $k$-tensor on $V$ as $k$-multilinear map on $V\times \cdots \times V$.
Question. I a wondering how could we define $^\sigma\alpha$ without thinking of tensors as multilinear maps.
A $k$-tensor on $V$ can be defined using the universal property of tensor products with no reference to multilinear maps. I am looking for a similar description of the action of $S_k$ on $\mathcal T^k(V^*)$.
Can somebody help?
Thanks.