Symmetric Tensor - Definition

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