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permutation operator (Definition)

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

$\displaystyle \phi ( v_1, \ldots , v_n ) = v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(n)}. $
Then by the universal factorization property for a tensor product there is a unique linear map $ P(\sigma) : V^{\otimes n} \to V^{\otimes n}$ such that $ P(\sigma)\otimes = \phi$. Then of course,
$\displaystyle 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$.

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})$



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Cross-references: nonsingular, identity mapping, linear map, map, multilinear, symmetric group, field, vector space
There are 2 references to this entry.

This is version 4 of permutation operator, born on 2006-09-16, modified 2006-09-16.
Object id is 8368, canonical name is PermutationOperator.
Accessed 875 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
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