# $\sigma(gv_1\otimes\ldots\otimes gv_n)=gv_{\sigma^{-1}(1)}\otimes\ldots\otimes gv_{\sigma^{-1}(n)}$?

Let $$V$$ be a $$\mathbb{C}G$$-module, then $$V^{\otimes n}$$ can be endowed with a $$\mathbb{C}G$$-module structure via $$g(v_1\otimes\ldots\otimes v_n)=gv_1\otimes \ldots \otimes gv_n$$. Now consider $$Sym^n(V)=\{w\in V^{\otimes n}| \sigma(w)=w \forall \sigma \in S_n\}$$, where $$\sigma(w_1\otimes \ldots \otimes w_n)= w_{\sigma^{-1}(1)}\otimes\ldots \otimes w_{\sigma^{-1}(n)}$$. I have to show that $$Sym^n(V)$$ is a submodule. I can almost show it but I am missing the argument that says that $$\sigma(gv_1\otimes\ldots\otimes gv_n)=gv_{\sigma^{-1}(1)}\otimes\ldots\otimes gv_{\sigma^{-1}(n)}$$.

• Let $w_i=gv_i$. Then apply the definition of action of $\sigma$ on $V^{\otimes n}$. – richrow Aug 1 at 13:26
• Oh thanks! I could finish now – roi_saumon Aug 1 at 13:38