# Permutation group product

Let $$E,F$$ be two vector spaces and $$\varphi: \overbrace{E\times \cdots \times E}^{\text{p times}} \to F$$ a $$p$$-linear map. If $$\sigma$$ is a permutation on $$S_{p}$$, then we can define another $$p$$-linear map $$\sigma \varphi: \overbrace{E\times \cdots \times E}^{\text{p times}}\to F$$ by: $$(\sigma \varphi)(x_{1},...,x_{p}) := \varphi(x_{\sigma(1)},...,x_{\sigma(p)})$$

Now, my book says that if $$\tau, \sigma$$ are two permutations, then: $$(\tau \sigma)\varphi = \tau(\sigma \varphi)$$ However, according to my calculations: $$[(\tau \sigma)\varphi](x_{1},...,x_{p}) = \varphi(x_{\tau(\sigma(1))},...,x_{\tau(\sigma(p))}) = [\sigma(\tau \varphi)](x_{1},...,x_{p})$$ since $$[\sigma(\tau\varphi)](x_{1},...,x_{p}) = (\tau\varphi)(x_{\sigma(1)},...,x_{\sigma(p)}) = \varphi(x_{\tau(\sigma(1))},...,x_{\tau(\sigma(p))})$$. Where is my mistake?

There is a mistake when you write

$$[(\tau \sigma)\varphi](x_{1},...,x_{p}) = \varphi(x_{\tau(\sigma(1))},...,x_{\tau(\sigma(p))}) = [\sigma(\tau \varphi)](x_{1},...,x_{p})$$

You have

\begin{aligned}(\tau \sigma)\varphi(x_{1},...,x_{p}) &= \varphi(x_{(\tau\sigma)(1)},...,x_{(\tau\sigma)(p))})\\ &= \varphi(x_{(\tau\circ \sigma)(1)},...,x_{(\tau\circ \sigma)(p))})\\ &= \varphi(x_{(\tau(\sigma(1))},...,x_{(\tau(\sigma(p))})\\ &= \tau(\varphi(x_{\sigma(1)},...,x_{\sigma(p)})\\ &= \tau(\sigma\varphi(x_1,...,x_p)\\ \end{aligned}

Therefore the equality of your book $$(\tau \sigma)\varphi = \tau(\sigma \varphi)$$

In general it is not true that $$\varphi(x_{\tau(\sigma(1))},...,x_{\tau(\sigma(p))}) = [\sigma(\tau \varphi)](x_{1},...,x_{p})$$ -- this only applies if $$\tau$$ and $$\sigma$$ commute in $$S_p$$. Take for example $$p=3$$, $$\tau = (1\ 3), \sigma = (1\ 2\ 3)$$. Then $$\sigma\tau \phi = (1\ 2\ 3)(1\ 3)\phi = \phi(x_1,x_3,x_2)$$ $$\tau\sigma \phi = (1\ 3)(1\ 2\ 3)\phi = \phi(x_2, x_1, x_3)$$

What is true is that when evaluating $$\sigma\tau\phi$$ (for example), you can first apply $$\tau$$ to $$\phi$$ and then finally apply $$\sigma$$, or you can first apply $$\sigma$$ to $$\tau$$ and then apply this combined permutation ($$\sigma\tau$$) to $$\phi$$.