Definition
Let V be a vector space and
a tensor of order r. Then T is a symmetric tensor if
for the braiding maps associated to every permutation σ on the symbols {1,2,...,r} (or equivalently for every transposition on these symbols).
Given a basis {ei} of V, any symmetric tensor T of rank r can be written as
for some unique list of coefficients (the components of the tensor in the basis) that are symmetric on the indices. That is to say
for every permutation σ.
The space of all symmetric tensors of rank r defined on V is often denoted by Sr(V) or Symr(V). It is itself a vector space, and if V has dimension N then the dimension of Symr(V) is the binomial coefficient
Read more about this topic: Symmetric Tensor
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