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symmetric tensor (Definition)

Let $ V$ be a vector space over a field. Let $ S_n$ be the symmetric group on $ \{1, \ldots, n\}$. An order-n tensor $ A \in V^{\otimes n}$ is said to be symmetric if $ P(\sigma)A = A$ for all $ \sigma \in S_n$, where $ P(\sigma)$ is the permutation operator associated to $ \sigma$. The set of symmetric tensors in $ V^{\otimes n}$ is denoted by $ S^n(V)$.



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Cross-references: permutation operator, symmetric group, field, vector space
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This is version 2 of symmetric tensor, born on 2006-09-16, modified 2006-09-16.
Object id is 8369, canonical name is SymmetricTensor.
Accessed 909 times total.

Classification:
AMS MSC: 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
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