$S_n$ acts on $V^{\otimes n}$ ($V$: a vector space) from the right as $$x_1\otimes ...\otimes x_n.\sigma=x_{\sigma(1)}\otimes ...\otimes x_{\sigma(n)},$$ but I don't see why as I get $$x_1\otimes ...\otimes x_n.(\sigma\circ\tau)=x_{\sigma(\tau(1))}\otimes ...\otimes x_{\sigma(\tau(n))}$$ which is not equal to $$(x_1\otimes ...\otimes x_n.\sigma).\tau=x_{\sigma(1)}\otimes ...\otimes x_{\sigma(n)}.\tau=x_{\tau(\sigma(1))}\otimes...\otimes x_{\tau(\sigma(n))}.$$
So what am I doing wrong here?
To avoid confusion on how $\tau$ acts on $(x_1\otimes ...\otimes x_n.\sigma)$, you could rename $v_i = x_{\sigma(i)}$ and notice that $(v_1\otimes ...\otimes v_n.\tau)=v_{\tau(1)}\otimes ...\otimes v_{\tau(n)} = x_{\sigma(\tau(1))}\otimes ...\otimes x_{\sigma(\tau(n))}$.