# permutation operator

Let $V$ be a vector space over a field. Let $\sigma\in S_{n}$, the symmetric group on $\{1,\ldots,n\}$ and define a multilinear map $\phi:V\times\cdots\times V\to V^{\otimes n}=\overbrace{V\otimes\cdots\otimes V% }^{n\text{ times}}$ by

 $\phi(v_{1},\ldots,v_{n})=v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}% (n)}.$

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

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

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

## 1 Properties

1. 1.

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

2. 2.

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

3. 3.

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

Title permutation operator PermutationOperator 2013-03-22 16:15:38 2013-03-22 16:15:38 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 15A04