permutation operator

Let $V$ be a vector space over a field. Let $\sigma \in S_n$, the symmetric group on $\{1,\mathrm{\dots },n\}$ and definea multilinear map $\varphi :V\times \mathrm{\cdots }\times V\to V^{\otimes n}=\stackrel{n\text{ times}}{\overbrace{V\otimes \mathrm{\cdots }\otimes V}}$ by

$$\varphi (v_1,\mathrm{\dots },v_n)=v_{{\sigma }^{-1}(1)}\otimes \mathrm{\cdots }\otimes v_{{\sigma }^{-1}(n)}.$$

Then by the universal factorization property (http://planetmath.org/TensorProduct) for a tensor product (http://planetmath.org/TensorProduct) there is aunique linear map $P(\sigma ):V^{\otimes n}\to V^{\otimes n}$ such that$P(\sigma )\otimes =\varphi$. Then of course,

$$P(\sigma )v_1\otimes \mathrm{\cdots }\otimes v_n=v_{{\sigma }^{-1}(1)}\otimes \mathrm{\cdots }\otimes v_{{\sigma }^{-1}(n)}.$$

$P(\sigma )$ is called the permutation operator associated with $\sigma$.

1 Properties

1. $P(\sigma \tau )=P(\sigma )P(\tau )$

2. $P(e)=I$ , where $I$ is the identity mapping on $V^{\otimes n}$

3. $P(\sigma )$ is nonsingular and $P{(\sigma )}^{-1}=P({\sigma }^{-1})$